A solid is bounded below by the cone, z=x2+y2, and bounded above by the sphere of radius 2 centered at the origin. Find integrals that compute its volume using Cartesian and cylindrical coordinates. For your answers use θ= theta.

Answer:

c) 44000 miles

Step-by-step explanation:

The regression line has the following format:

[tex]y = bx + a[/tex]

In which b is the slope(how much each mile costs) and a is the fixed number of miles flown.

In this problem, we have that:

[tex]a = 24000, b = 10[/tex]. So

[tex]y = 10x + 24000[/tex]

A person who spent $ 2000 is predicted to have flown:

This is y when x = 2000.

[tex]y = 10(2000) + 24000[/tex]

[tex]y = 44000[/tex]

So the correct answer is:

c) 44000 miles

To find the predicted distance flown by someone who spent $2000, you use the given intercept (a) and slope (b) in the straight line equation Y = a + bX. When $2000 is substituted for X in the equation, your solution is 44,000 miles.

In this problem, you are asked to determine the distance a person is predicted to have flown, given that the person spent $2000. Based on our regression line information, we know that the intercept (a) is 24,000 and the slope (b) is 10. Therefore, we can use the equation of the straight line Y = a + bX, where X represents the amount of money spent, and Y is the distance traveled. If we substitute $2000 for X in our equation, we find that Y = 24000 + 10 * 2000 = 44,000 miles. Therefore, the correct answer is option c) 44000 miles.

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Answer:

12%

Step-by-step explanation:

let x represent the actual number of members last year.

current members = 675 and with an increment of 13% over previous year

to find the increment

(675 - x) / x = 0.13

675 - x = 0.13x

collect the like terms

675 = 0.13x + x

675 = 1.13 x

x = approx 597 members were there last year

its hope to increase by the same number next year

percent increase = 675 - 597 / 675 = 78 / 675 = 0.12 = 12%

Answer:

($3.61, $4.27) is the range in which at least 88.9% of the data will reside.

Step-by-step explanation:

Chebyshev's Theorem

[tex]\mu \pm 2\sigma\\\\1 - \dfrac{1}{(2)^2}\% = 75\%[/tex]

Atleast 75% of data lies within two standard deviation of the mean for a non-normal data.

[tex]\mu \pm 3\sigma\\\\1 - \dfrac{1}{(3)^2}\% = 88.9\%[/tex]

Thus, range of data for which 88.9% of data will reside is

[tex]\mu \pm 3\sigma\\= (\mu - 3\sigma , \mu + \3\sigma)[/tex]

Now,

Mean = $3.94

Standard Deviation = $0.11

Putting the values, we get,

Range =

[tex]3.94 \pm 30.11\\= (3.94 - 3(0.11) , 3.94 + 3(0.11))\\=(3.61,4.27)[/tex]

($3.61, $4.27) is the range in which at least 88.9% of the data will reside.

Answer:

Using Chebyshev's Theorem, the range in which at least 88.9% of the data reside is [$3.82, $4.06].

Step-by-step explanation:

Given:

The mean of the prices of a gallon of milk is, [tex]\mu=\$3.94[/tex]

The standard deviation of the prices of a gallon of milk is, [tex]\sigma=\$0.11[/tex]

The Chebyshev's Theorem states that for a random variable X with finite mean μ and finite standard deviation σ and for a positive constant k the provided inequality exists,

                                      [tex]P(|X-\mu|\geq k)\leq \frac{\sigma^{2}}{k^{2}}[/tex]

The value of [tex]\frac{\sigma^{2}}{k^{2}} = 0.889[/tex]

Then solve for k as follows:

[tex]\frac{\sigma^{2}}{k^{2}} = 0.889\\\frac{(0.11)^{2}}{k^{2}}=0.889\\k^{2}=\frac{(0.11)^{2}}{0.889}\\k=\sqrt{0.0136108} \\\approx0.1167[/tex]

The range in which at least 88.9% of the data will reside is:

[tex]P(|X-\mu|\geq k)\leq \frac{\sigma^{2}}{k^{2}}\\P(\mu-k\leq X\leq \mu+k)\leq 0.889\\P(3.94-0.1167\leq \leq X\leq 3.94+0.1167)\leq 0.889\\P(3.8233\leq X\leq 4.0567)\leq 0.889\\\approxP(3.82\leq X\leq 4.06)\leq 0.889[/tex]

Thus, the probability of prices of a gallon of milk between $3.82 and $4.06 is 0.889.

Answer:

[tex]T(t) = 42000 + 21t[/tex]

Step-by-step explanation:

There are two costs for the operation of the equipment, the purchase cost and the hourly cost. The total cost is the sum of these two costs:

So

[tex]T = F_{c} + H_{c}[/tex]

In which [tex]F_{c}[/tex] is the fixed cost and [tex]H_{c}[/tex] is the hourly cost.

Fixed cost

A roofing contractor purchases a shingle delivery truck with a shingle elevator for $42,000. This means that the fixed cost is 42000. So [tex]F_{c} = 42000[/tex]

Hourly cost

The vehicle requires an average expenditure of $9.50 per hour for fuel and maintenance, and the operator is paid $11.50 per hour. So for each hour, there are expenses of 9.5 + 11.5 = $21.

So the hour cost is

[tex]H_{c}(t) = 21t[/tex]

In which t is the number of hours

Total cost

[tex]T = F_{c} + H_{c}[/tex]

[tex]T(t) = 42000 + 21t[/tex]

Answer:4000 bottle containers and 1000

Medical kits

Step-by-step explanation:

The total weights the plane can take per trip is 90000lb. It implies that if we multiply the weight of each bottle container by the number of bottle container and add it to the product of the number medical kit and the weight of each medical kit, we will obtain a total weight of 90000lb.

If x is the number of bottle and y is the number of medical kit

20×x+10×y=90000....i

Also

The total volume the plane can take per trip is 6000ft³ It implies that if we multiply the volume of each bottle container by the number of bottle container and add it to the product of the number medical kit and the volume of each medical kit, we will obtain a total volume of 6000ft³.

Recall, x is the number of bottle and y is the number of medical kit

1×x+2×y=6000....ii

Combining equation i and ii and solving simultaneously,

x=4000 units and y= 1000 units in each plane trip

Answer:

Bottled water = 4000.

Medical kits = 1000.

Step-by-step explanation:

Let x = bottled water

y = medical kits

For the mass (in pounds):

20x + 10y = 90000

For the cubic ft:

x + 2y = 6000

Solving equation i and ii simultaneously,

x = 4000.

y = 1000.

Bottled water = 4000.

Medical kits = 1000.

Answer:

a) 0.0989, Option a

Step-by-step explanation:

The concept of Poisson probability distribution is used as shown in the attached file.

Answer:

[tex]45*10^{6}[/tex] orders of magnitude.

Step-by-step explanation:

We know that 900 km is equivalent to 20 cm, because we need to fit the United Kingdom length on a sheet of paper. So:

Therefore, if we divide 45 km by 0.00001 km (1 cm) we will have [tex]45*10^{6}[/tex] times to scale down.

I hope it helps you!

Answer:

the helix intersects the sphere at t=4 and t=(-4)

Step-by-step explanation:

for the helix r(t) = [ sin(t) , cos(t) ,  t ] then x=sin(t) , y=cos(t) and z=t

thus the helix intersect the sphere  x² + y² + z² = 17 at

x² + y² + z² = 17

[sin(t)]²+[cos(t)]²+ t² = 17

1 + t² = 17

t² = 16

t = ±4

thus the helix intersects the sphere at t=4 and t=(-4)

The point wher the helix intersect the sphere is at t =  ±4

Given the coordinate of a helix expressed as;

If these coordinate intersects the sphere  x² + y² + z² = 17

Substitute the coordinate of the helix:

x² + y² + z² = 17

(sint)² + (cost)² + t² = 17

sin²t + cos²t + t² = 17

1 + t² = 17

t² = 17 - 1

t² = 16

t = ±√16

t = ±4

Hence the point wher the helix intersect the sphere is at t =  ±4

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Answer:

(a) 0.970225.

(b) 0.999775.

(c) 2.25 x [tex]10^{-4}[/tex].

Step-by-step explanation:

Given the probability that either system will function satisfactorily during a flight is 0.985.

(a) To calculate the probability that both systems function satisfactorily we are given the probability of either system will function of 0.985 which means that both function have probability of functioning satisfactorily of 0.985 each i.e.,

= 0.985 x 0.985 = 0.970225.

(b) The probability of first function will not function satisfactorily is 1 - 0.985 because the probability of first system function satisfactorily is 0.985.

Similarly, probability of second function will not function satisfactorily is also 1 - 0.985 = 0.015

So Probability that during a given flight at least one system functions satisfactorily = 1 - both system will not function satisfactorily

                     = 1 - (0.015 x 0.015) = 0.999775.

(c) Probability that during a given flight both systems fail or not function  satisfactorily = (1 - 0.985) x (1 - 0.985) = 2.25 x [tex]10^{-4}[/tex]

Answer:

(a) P (Both systems functioning satisfactorily) = 0.970

(b) P (At least one system functions satisfactorily) = 0.999

(c) P (Both the systems failing) = 0.00023

Step-by-step explanation:

Let A = the system in use is working satisfactorily and B = the backup system is working satisfactorily.

The probability that either of the systems works satisfactorily is,

P (A) = P(B) = 0.985

(a)

Both the events A and B are independent, i.e. [tex]P(A\cap B)=P(A)\times P(B)[/tex]

Compute the probability that during a given flight both systems function satisfactorily as follows:

[tex]P(A\cap B)=P(A)\times P(B)[/tex]

               [tex]=0.985\times0.985\\=0.970225\\\approx0.970[/tex]

Thus, the probability of both systems functioning satisfactorily is 0.970.

(b)

Compute the probability that during a given flight at least one system functions satisfactorily as follows:

P (At least one system functions satisfactorily) = 1 - P (None of the system functions satisfactorily)

                                                                             = [tex]1-[P(A^{c})\times P(B^{c})][/tex]

                                                                             [tex]=1-([1-P(A)]\times [1-P(B)])\\=1-([1-0.985]\times [1-0.985])\\= 1-0.000225\\=0.999775\\\approx0.999[/tex]

Thus, the probability that during a given flight at least one system functions satisfactorily is 0.999.

(c)

Compute the probability that during a given flight both the systems fail as follows:

P (Both the systems failing) = [tex]P(A^{c})\times P(B^{c})[/tex]

                                             [tex]=[1-P(A^{c})]\times [1-P(B^{c})]\\=(1-0.985)\times (1-0.985)\\=0.015\times 0.015\\=0.000225\\\approx0.00023[/tex]

Thus, the probability that during a given flight both the systems fail is 0.00023.

Answer:

A. 8 complete hotdogs were made.

B. The wiener.

C. The bun.

Step-by-step explanation:

1 complete hotdog = 1 wiener + 1 bun

You have, 3*10 bun and 8 wiener

= 30 bun and 8 wiener

A. Since, we have 30 bun and 8 wiener and for 1 complete hotdog = 1 wiener + 1 bun

Therefore, 8 hotdogs = 8 wiener + 8 buns

= 8 complete hotdogs are made

B.

Since 8 complete hotdogs are made, 8 wiener and 8 buns are used.

Therefore, the limiting ingredient is the wiener because it is the smallest number of ingredient and none of it was left after making the 8 hotdogs.

C. Since 8 complete hotdogs are made, 8 buns were used

So, the leftover buns = (30 - 8) buns

= 22 buns

The buns is the leftover ingredient

At the store, you buy four packages of eight wieners and three bags of 10 buns, you can make 8 hotdogs.

There are 8 hotdogs you can make then the limited ingredient is wieners.

The buns are the leftover ingredient.

You and your friends are making hot dogs.

A complete hot dog consists of a wiener and a bun.

At the store, you buy four packages of eight wieners and three bags of 10 buns.

Total number of buns = 30

Total number of wieners = 8

At the store, you buy four packages of eight wieners and three bags of 10 buns.

For every hot dog, 1 wiener one 1 buns are required. the calculation for the given question is given as follows;

1. The total number of hot dogs can you make is,

To make 1 hot dog 1 wiener and 1 bun

Then,

for 8 hot dogs, 8 wieners and 8 buns are required.

Therefore, you can make 8 hotdogs.

2. There are 8 hotdogs you can make then the limited ingredient is wieners.

3. After making 8 hot dogs only 8 buns are used,

Then,

Left buns = Total number of buns - used buns

Left buns = 30-8 = 22

The buns are the leftover ingredient.

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Answer:

See below

Step-by-step explanation:

Essentially, we have to replace "quantifier words" like "All", "Every" by the universal quantifier ∀.

a) ∀ dinosaur x, x is extinct.

b) ∀ real number x, x is positive, negative, or zero.

c) ∀ irrational number x, x is not an integer.

d) ∀ logician x, x is not lazy.

e) ∀ integer x, x²≠ 2,147,581,953.

f)  ∀ real number x, x²≠ -1.

In a) and b) we replace the words without major changes. In the other statements, we modify the statement using negation. For example, "No irrational numbers are integers." is equivalent to "Every irrational number is not integer".

Answer:

Step-by-step explanation:

The degree of a polynomial is the sum of the exponents of all of its variables. Where there is only one variable, the degree would be the highest power to which that variable is raised. The terms are the individual components that make up the polynomial. Therefore,

16t2 is a second degree because the variable, t is raised to 2 and the term is 1

240 is zero degree and 1 term.

15,000 – 352t is first degree and 2 terms.

The total number of ways to choose the two groups, depending on the relationship between n and m, is:

M < N: C(100, n).

M ≥ N: Σ (C(100, k) * C(100 - k, m - k)) for k = n to 1.

Case 1: m < n (n tallest chosen first):

Choose the n tallest people from the 100. There are C(100, n) ways to do this.

The remaining 100 - n people are all shorter than the chosen n. Since m < n, we can simply choose all the remaining people (100 - n).

The total number of ways in this case is C(100, n).

Case 2: m ≥ n (not all n tallest need to be chosen):

This case requires considering all possible combinations of how many tall people to choose (from n down to 1).

For each value of k (n, n - 1, ..., 1), choose k tallest people from the 100. There are C(100, k) ways to do this.

The remaining 100 - k people are all shorter than the chosen k. Choose the remaining m - k people from this group. There are C(100 - k, m - k) ways to do this.

Sum the results for each value of k: Σ (C(100, k) * C(100 - k, m - k)) for k = n to 1.

Therefore, the total number of ways to choose the two groups, depending on the relationship between n and m, is:

M < N: C(100, n)

M ≥ N: Σ (C(100, k) * C(100 - k, m - k)) for k = n to 1

Complete question:

How many ways are there to pick a group of n people from 100 people, and then pick a second group of m people from the remaining 100 − n people, so that all the people you chose from the first group are taller than the people from the second group (assume that everyone in the group of 100 people has a different height.)

Note: it is implied in this question that 1 ≤ n ≤ 100 and 0 ≤ m ≤ 100 − n.

Answer: There is a strong positive correlation between  number of games won by a minor league baseball team and the average attendance at their home games is analyzed.

Step-by-step explanation:

The Pearson's coefficient 'r' gives the correlation between the predicted values and the observed values .

Given :  A regression to predict the average attendance from the number of games won has an r = 0.73.

Since r=0.73 is positive and 0.70 <0.73 <1 , it means there is a strong positive correlation between number of games won by a minor league baseball team and the average attendance at their home games is analyzed.

Answer:

The number of possible choices of my team and the opponents team is

 [tex]\left\begin{array}{ccc}n-1\\E\\n=1\end{array}\right i^{3}[/tex]

Step-by-step explanation:

selecting the first team from n people we have [tex]\left(\begin{array}{ccc}n\\1\\\end{array}\right) = n[/tex] possibility and choosing second team from the rest of n-1 people we have [tex]\left(\begin{array}{ccc}n-1\\1\\\end{array}\right) = n-1[/tex]

As { A, B} = {B , A}

Therefore, the total possibility is [tex]\frac{n(n-1)}{2}[/tex]

Since our choices are allowed to overlap, the second team is [tex]\frac{n(n-1)}{2}[/tex]

Possibility of choosing both teams will be

[tex]\frac{n(n-1)}{2} * \frac{n(n-1)}{2} \\\\= [\frac{n(n-1)}{2}] ^{2}[/tex]

We now have the formula

1³ + 2³ + ........... + n³ =[tex][\frac{n(n+1)}{2}] ^{2}[/tex]

1³ + 2³ + ............ + (n-1)³ = [tex][x^{2} \frac{n(n-1)}{2}] ^{2}[/tex]

=[tex]\left[\begin{array}{ccc}n-1\\E\\i=1\end{array}\right] = [\frac{n(n-1)}{2}]^{3}[/tex]

Answer:

15 is the number of combination of 4 successes.

Step-by-step explanation:

We are given the following information:

We are given a binomial distribution, then probability of x succes is given by

[tex]P(X=x) = \binom{n}{x}.p^x.(1-p)^{n-x}[/tex]

where n is the total number of observations, x is the number of success, p is the probability of success.

Now, we are given n = 6 and x = 4

We have to evaluate the number of combination of success and failures.

It is given by:

[tex]\binom{n}{x} = \dfrac{n!}{x!(n-x)!}\\\\\binom{6}{4} = \dfrac{6!}{4!(6-4)!} = \dfrac{6!}{4!2!} = 15[/tex]

Thus, 15 is the number of combination of 4 successes.

The number of combinations of 4 successes out of 6 attempts, calculated using the binomial probability combination formula, is 15.

In binomial probability, the number of ways to get a specific number of successes is often calculated using the combination formulas. In this case, the number of combinations of X successes would be determined by the combination formula C(n, x), which stands for the number of combinations of n items taken x at a time. In your case, where n=6 and x=4, the required number of combinations (or ways to achieve 4 successes out of 6 trials) is C(6,4).

The formula for a combination is generally given by: n! / [x!(n - x)!]. Plugging the given numbers into the formula, we get: C(6,4) = 6! / [4!(6 - 4)!].  This simplifies to: 720 / (24*2), which equals to 15. So, there are 15 combinations of 4 successes out of 6 attempts.

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Answer:

0.375 feet-lb

Step-by-step explanation:

We have been given that the work required to stretch a spring 2 ft beyond its natural length is 6 ft-lb. We are asked to find the work needed to stretch the spring 6 in. beyond its natural length.

We can represent our given information as:

[tex]6=\int\limits^2_0 {F(x)} \, dx[/tex]

We will use Hooke's Law to solve our given problem.

[tex]F(x)=kx[/tex]

Substituting this value in our integral, we will get:

[tex]6=\int\limits^2_0 {kx} \, dx[/tex]

Using power rule, we will get:

[tex]6=\left[ \frac{kx^2}{2} \right ]^2_0[/tex]

[tex]6=\frac{k(2)^2}{2}-\frac{k(0)^2}{2}[/tex]

[tex]6=\frac{4k}{2}-0\\\\k=3[/tex]

We know that 6 inches is equal to 0.5 feet.

Work needed to stretch it beyond 6 inches beyond its natural length would be [tex]\int\limits^{0.5}_0 {kx} \, dx =\int\limits^{0.5}_0 {3x} \, dx[/tex]

Using power rule, we will get:

[tex]\int\limits^{0.5}_0 {3x} \, dx = \left [\frac{3x^2}{2}\right]^{0.5}_0[/tex]

[tex]\frac{3(0.5)^2}{2}-\frac{3(0)^2}{2}\Rightarrow \frac{3(0.25)}{2}-0=\frac{0.75}{2}=0.375[/tex]

Therefore, 0.375 feet-lb work is needed to stretch it 6 in. beyond its natural length.

Answer:

Step-by-step explanation:

Total mass of water = density X volume = 1000 kg/m3 x 187.5 m3

where H2 = initial position of centre of mass from ground

where H1 = final position of centre of mass from ground

The work required to pump the water out of the trough is [tex]5.11\times 10^6[/tex] J and this can be determine by using the formula of work done.

Given :

a) The work required to pump the water out of the trough is given by:

[tex]\rm W=mg(H_2-H_1)[/tex]    --- (1)

where m is the total mass of the water [tex]\rm H_1[/tex] is the initial position and [tex]\rm H_2[/tex] is the final position of the center of mass.

So. the mass of the water is given below:

[tex]=1000\times 187.5[/tex]

= 187500 Kg

Now, the value of the initial position of the center of mass is zero and the value of the final position of the center of mass is 2.78 m.

Now, substitute the known terms in the expression (1).

[tex]\rm W = 187500\times 9.8\times (2.78-0)[/tex]

[tex]\rm W = 5.11\times 10^{6}\;J[/tex]

b) The width of the cylinder is given by:

[tex]\rm w-5=\dfrac{5-2.5}{0-5}(x-0)[/tex]

Simplify the above expression.

w = -0.5x + 5

Now, the volume of the cylinder is given by:

[tex]\rm dv = 10wdx[/tex]

Substitute the value of w in the above expression.

[tex]\rm dv = 10(-0.5x+5)dx[/tex]

dv = (50-5x) dx

Now, the value of the force is given by:

[tex]\rm dF = 9800dv[/tex]

Substitute the value of dv in the above expression.

dF = 9800(50 - 5x) dx

Now, the expression of work is given by:

dW = x dF

Substitute the value of force in the above expression.

dW = x(9800(50 - 5x)) dx

Now, integrate the above expression.

[tex]\rm W = 9800\int^5_0(50x-5x^2)dx[/tex]

[tex]\rm W = 9800 \times \left(25x^2-\dfrac{5x^3}{3}\right)^5_0[/tex]

[tex]\rm W= 9800\left(625-\dfrac{625}{3} \right)[/tex]

W = 4083333.34 J

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Answer: a) C = 0.126x + 1.5

b) C = 2.382 c) approximately 179 units

Step-by-step explanation: profit P(x), total revenue R(x) and total cost (C) are related by the formulae below.

Profit = total revenue - total cost

P(x) = R(x) - C

From the question, P(x) = 0.084x - 1.5, R(x) = 0.21x.

Question a)

C = R(x) - P(x)

C = 0.21x - {0.084x - 1.5}

C = 0.21x - 0.084x + 1.5

C = 0.126x + 1.5

Question b)

If C = 0.126x + 1.5, then C at x = 7 units is gotten below as

C = 0.126(7) + 1.5

C = 0.882 + 1.5

C = 2.382.

Question c)

The break even point is the point where total revenue R(x) equals total cost C.

R(x) = 0.21x and C = 0.126x + 1.5

0.21x = 0.126x + 1.5

0.21x - 0.126x = 1.5

0.084x = 15

x = 15/ 0.084

x = 178.57 which is approximately 179 units (since quantity of units can't be decimal)

To find the cost equation, subtract the profit equation from the revenue equation. The cost of producing 7 units is approximately 2.382 million dollars. The break-even point occurs at approximately 17.857 units.

a. To find the cost equation, we need to subtract the profit equation from the revenue equation. Since the revenue equation is R(x) = .21x and the profit equation is P(x) = .084x - 1.5, the cost equation is C(x) = R(x) - P(x). Substituting the given equations, we have C(x) = .21x - (.084x - 1.5).

Simplifying the equation, we get C(x) = .21x - .084x + 1.5.

Combining like terms, the cost equation is C(x) = .126x + 1.5.

b.

To find the cost of producing 7 units, we can substitute x = 7 into the cost equation. C(7) = .126(7) + 1.5 = .882 + 1.5 = 2.382 million dollars.

c.

The break-even point occurs when the revenue equals the cost. To find the break-even point, we set R(x) = C(x). Substituting the given equations, we have .21x = .126x + 1.5.

Solving for x, we get .084x = 1.5.

Dividing both sides by .084, we find x = 17.857 units.

Therefore, the break-even point is approximately 17.857 units.

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Answer:

the probability is 0.311 (31.1%)

Step-by-step explanation:

defining the event L= being late to work :Then knowing that each mode of transportation is equally likely (since we do not know its travel habits) :

P(L)= probability of taking the bicycle * probability of being late if he takes the bicycle + probability of taking the car* probability of being late if he takes the car  + probability of taking the bus* probability of being late if he takes the bus +probability of taking the train* probability of being late if he takes the train = 1/4  * 0.75 +   1/4 * 0.43 + 1/4  * 0.15  + 1/4 * 0.05 = 0.345

then we can use the theorem of Bayes for conditional probability. Thus defining the event C= Bob takes the car , we have

P(C/L)=  P(C∩L)/P(L) = 1/4 * 0.43 /0.345 = 0.311 (31.1%)

where

P(C∩L)= probability of taking the car and being late

P(C/L)= probability that Bob had taken the car given that he is late

If Bob is late without considering the transportation mode, the total probability is 63%. Using Bayes' theorem, we find the probability that Bob was late because he took the car is approximately 17%.

Firstly, it's essential to ascertain the total chances of Bob being late disregarding the mode of transport. And that is the sum of his possibilities of being late for each mode, which is 43% for the car, 15% for the bus, and 5% for the train: 43% + 15% + 5% = 63%. Thus, if Bob is late, there's a 63% chance that he took either car, bus, or train. To calculate the probability that Bob took the car given that he was late, we employ Bayes' Theorem.

The formula for Bayes' theorem is P(A|B) = P(B|A)P(A) / P(B). In this instance, event A is Bob travelling by car, event B is Bob being late. We want to estimate P(A|B), the chance of A happening given B has occurred. Hence we plug in values: P(A|B) = (0.43)(0.25) / (0.63) ≈ 0.17 or 17%. Thus, the boss would estimate that there's a 17% chance that Bob took his car if he observes Bob coming late.

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Answer:

We conclude that its value change for 277.8$.

Step-by-step explanation:

We know that a perpetuity pays $50 per year and interest rates are 9 percent.  We calculate how much would its value change if interest rates decreased to 6 percent. We know that

9%=0.09

6%=0.06

We get

\frac{50}{0.09}=555.5

\frac{50}{0.06}=833.3

Therefore, we get 833.3-555.5=277.8

We conclude that its value change for 277.8$.

The value change for $277.8.

[tex]\frac{50}{0.09}=555.5\\\\\frac{50}{0.06}=833.3[/tex]

So,

=  $833.3  - $555.5

= $277.8

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The speed at which each bicyclist rides is 320 times the distance covered by each bicyclist.

To solve this problem, we can consider the distance traveled by the train and by each bicyclist. Let's assume the distance covered by each bicyclist is x miles. The train covers the remaining 1 - 7/8 = 1/8 mile. We can set up an equation to represent the time it takes for the train and the bicyclists to travel their respective distances:

Time taken by train = distance traveled by train / speed of train = (1/8) mile / 40 mph = 1/320 hour

Time taken by each bicyclist = distance traveled by bicyclist / speed of bicyclist = x miles / v mph, where v represents the speed of the bicyclists

Since the train passes each bicyclist when they are 7/8 of the way through the tunnel, the time it takes for each bicyclist to travel their distance is equal to the time it takes for the train to travel its distance:

(x / v) = 1/320

To solve for v, we can rearrange the equation:

v = x / (1/320)

v = 320x mph

The speed at which each bicyclist rides is 320 times the distance covered by each bicyclist.

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Final answer:

To escape a train traveling at 40 mph through a tunnel, two bicyclists traveling in opposite directions must ride at a speed of exactly 35 mph to escape the tunnel safely. We derive this by setting up two inequalities based on the distances each cyclist must cover and then finding the minimum speed at which these conditions are satisfied.

Explanation:

We are discussing a classic algebra problem that utilizes rate, time, and distance calculations. Two bicyclists are 7/8 of the way through a mile-long tunnel when a train traveling at 40 mph approaches from behind. Each cyclist pedals at the same speed but in opposite directions to escape the tunnel as the train passes by them.

To solve the problem, we need to determine how far each bicyclist has to travel to escape the tunnel. One bicyclist has 1/8 mile to exit, while the other bicyclist has 7/8 mile. Since they travel at the same speed and need to escape before the train reaches them, we can set up a ratio based on distances and speeds. We know the train covers the mile in (1/40) hours since speed is distance over time.

The bicyclist closer to the exit (1/8 mile away) must bike this distance faster than the train covers the entire mile to avoid collision, and similarly for the second bicyclist (7/8 miles away). Let's assume their biking speed is 's'. Then, their times to escape are (1/8)/s and (7/8)/s respectively. These times must be less or equal to the time it takes for the train to travel the mile, which is 1/40 hours.

Setting up the first inequality: (1/8)/s ≤ 1/40
Which simplifies to: s ≥ 5 mph
And the second inequality: (7/8)/s ≤ 1/40
This second inequality simplifies to: s ≥ 35 mph
Since both conditions must be satisfied and s cannot be greater than 35 mph for both to be true, the speed at which both bicyclists must ride to escape is exactly 35 mph.

Answer:

a. 15

b.

Sr.no Samples Sample mean

1           (79,64)         71.5

2          (79,84)          81.5

3          (79,82)          80.5

4          (79,92)          85.5

5          (79,77)           78

6         (64,84)            74

7          (64,82)            73

8          (64,92)            78

9          (64,77)            70.5

10        (84,82)            83

11         (84,92)            88

12        (84,77)             80.5

13        (82,92)            87

14        (82,77)            79.5

15        (92,77)            84.5

c.

mean of sample mean=population mean=79.67

Step-by-step explanation:

a.

The different samples of two test grade are nCr, where n=6 and r=2.

nCr=6C2=6!/2!(6-2)!=6*5*4!/2!4!=30/2=15.

Thus, there are 15 different samples of two test grade.

b.

All possible samples are listed below:

Sr.no Samples

1           (79,64)        

2          (79,84)          

3          (79,82)          

4          (79,92)        

5          (79,77)          

6         (64,84)            

7          (64,82)

8          (64,92)

9          (64,77)

10        (84,82)

11         (84,92)

12        (84,77)

13        (82,92)

14        (82,77)

15        (92,77)

The sample means for each sample can be calculated as

Sr.no Samples Sample mean

1           (79,64)         (79+64)/2=71.5

2          (79,84)          (79+84)/2=81.5

3          (79,82)          (79+82)/2=80.5

4          (79,92)          (79+92)/2=85.5

5          (79,77)           (79+77)/2=78

6         (64,84)            (64+84)/2=74

7          (64,82)            (64+82)/2=73

8          (64,92)            (64+92)/2=78

9          (64,77)            (64+77)/2=70.5

10        (84,82)            (84+82)/2=83

11         (84,92)            (84+92)/2=88

12        (84,77)             (84+77)/2=80.5

13        (82,92)            (82+92)/2=87

14        (82,77)            (82+77)/2=79.5

15        (92,77)            (92+77)/2=84.5

c.

The sample means of sample mean μxbar will calculated by taking average of sample means

μxbar=(71.5+ 81.5+ 80.5+ 85.5+ 78+ 74+ 73+ 78+ 70.5+ 83+ 88+ 80.5+ 87+ 79.5+ 84.5)/15

μxbar=1195/15=79.67

Population mean=μ=(79+64+84+82+92+77)/6

μ=478/6=79.67

Sample means of sample mean μxbar=Population mean μ.

The missing value in the table is 15.

Solution:

Ratio of x to y coordinate:

[tex]$\frac{x}{y}= \frac{1}{3}[/tex]

[tex]$\frac{x}{y}= \frac{2}{6}= \frac{1}{3}[/tex]

[tex]$\frac{x}{y}= \frac{8}{24}= \frac{1}{3}[/tex]

So, the ratios are equivalent ratios.

To find the missing value in the table:

Multiply the x-coordinate by 3, we get the y-coordinate.

x-coordinate = 5

y-coordinate = x-coordinate × 3

                     = 5 × 3

                     = 15

Hence the missing value in the table is 15.

Answer:

15

Step-by-step explanation:

1 multiplied by 5 equals 5. Do the same with the y-value. 3 multiplied by 5 is 15!

Using it's concept, it is found that the proportion of Protestants that were calm for 2 out of the 3 days is of 0.0606.

In this problem:

Hence:

[tex]p = \frac{6}{99} = 0.0606[/tex]

The proportion of Protestants that were calm for 2 out of the 3 days is of 0.0606.

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Approximately 6.06% of Protestants were calm for 2 out of the 3 days, calculated by dividing the number of Protestants calm for 2 days (6) by the total surveyed (99).

To find the proportion of Protestants who were calm for 2 out of the 3 days, we need to look at the numbers given for Protestant individuals and calculate the ratio of those who were calm for exactly two days to the total number of Protestants surveyed. According to the data provided:

The total number of Protestants surveyed is the sum of those calm for 0, 1, 2, and 3 days, which is 50 + 27 + 6 + 16 = 99.
To find the proportion of Protestants calm for 2 days, we divide the number calm for 2 days by the total number of Protestants surveyed:

Proportion = Number calm for 2 days / Total number of Protestants
Proportion = 6 / 99
Proportion ≈ 0.0606

This means that approximately 6.06% of Protestants were calm for 2 out of the 3 days.

Answer and Step-by-step explanation:

a) probability of selling a medium, long sleeved printed shirt, P(M ∩LS ∩PR) = 0.07, directly from the table of probabilities

b) probability that the next shirt sold is a medium printed shirt, (M ∩Pr) = P(M,Pr,LS) + P(M,Pr,SS) = 0.07+0.10 = 0.17

c) probability that the next shirt is a short sleeved shirt, P(SS) = sum of 9 probabilities in Short Sleeved shirt table = 0.58

probability that the next shirt is a long sleeved shirt, P(LS) = 1 - 0.58 = 0.42 or add up the total probabilities in the long sleeved shirt table, P(LS) = sum of 9 probabilities in the long sleeved shirt table = 0.42

d) probability that the next shirt is a medium, P(M) = P(M,SS) + P(M,LS) = (0.07+0.10+0.12) + (0.06+0.07+0.07) = 0.49

probability that the next shirt sold is a print, P(Pr) = P(Pr,SS) + P(Pr,LS) = (0.02+0.10+0.07) + (0.02+0.07+0.02) = 0.40

e) probability that the shirt sold is a medium given that the shirt just sold was a short-sleeved plaid, P(M|SS,PL) = (P(M,SS,PL))/P(SS,PL) = 0.07/(0.04+0.07+0.03) = 0.5

f) probability that the shirt sold is short sleeved given that the shirt just sold was a medium plaid, P(SS|M,PL) = (P(M,SS,PL))/P(M,PL) = 0.07/(0.07+0.06) = 0.53846 = 0.54

probability that the shirt sold is long sleeved given that the shirt just sold was a medium plaid, P(LS|M,PL) = (P(M,LS,PL))/P(M,PL) = 0.06/(0.07+0.06) = 0.462 = 0.46

QED!

The following probabilities are as follow;

Probability means possibility. It deals with the occurrence of a random event. The value of probability can only be from 0 to 1. Its basic meaning is something is likely to happen. It is the ratio of the favorable event to the total number of events.

(a) The next shirt sold is a medium, long-sleeved, print shirt.

Probability = 0.07

(b) The probability that the next shirt sold is a medium print shirt.

Probability = 0.7 + 0.10 = 0.17

(c) The probability that the next shirt sold is a long-sleeved shirt.

Probability = 0.42

(d) The probability that the size of the next shirt sold is medium. The probability that the pattern of the next shirt sold is a print.

Probability = 0.07 + 0.10 = 0.17

(e)The shirt just sold was a short-sleeved plaid, the probability that its size was medium.

Probability = 0.07

(f) The shirt just sold was a medium plaid, The probability that it was short-sleeved.

Probability = 0.07

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Answer:

a) For this case we can see on the y axis the test scores and on the x axis the hours of homework. And as we can see we have a proportional relationship, because when the hours of homework increase the test scores increases too. If we fit a lineal model between y and x we will see an slope positive and a correlation coefficient positive based on the data observed.

b) [tex] y = 8x +42[/tex]

And we want to predict the test score for x = 3hr of homework we just need to replace the value of x=3 in the linear model and we got:

[tex] y = 8*3 +42= 24+42=66[/tex]

And that would be our predicted value for 3 h of homework

Step-by-step explanation:

For this case we consider the scatter plot attached to solve the problem.

Part a

For this case we can see on the y axis the test scores and on the x axis the hours of homework. And as we can see we have a proportional relationship, because when the hours of homework increase the test scores increases too. If we fit a lineal model between y and x we will see an slope positive and a correlation coefficient positive based on the data observed.

Part b

Assuming the following linear model for the situation:

[tex] y = 8x +42[/tex]

And we want to predict the test score for x = 3hr of homework we just need to replace the value of x=3 in the linear model and we got:

[tex] y = 8*3 +42= 24+42=66[/tex]

And that would be our predicted value for 3 h of homework

Answer:

1) P(E) = 1 - 0.14 = 0.86

2) P(E) = 1 - 0.92 = 0.08

3) P(E) = 1 - 17/35 = 18/35

4) P(E) = 1 - 61/100 = 39/100

Step-by-step explanation:

The probability that an event will happen equals one minus the probability that an event will not happen.

P(E) = 1 - P(E') ......1

Where;

P(E) is the probability that an event will happen.

P(E') is the probability that an event will not happen.

1. P(E') = 0.14

Applying equation 1

P(E) = 1 - 0.14 = 0.86

2. P(E') = 0.92

P(E) = 1 - 0.92 = 0.08

3. P(E') = 17/35

P(E) = 1 - 17/35 = 18/35

4. P(E') = 61/100

P(E) = 1 - 61/100 = 39/100

the last cow gives 202 pints of milk

Given :

There are 199 cows line up in the barn and he begins milking them.

The first cow gives 4 pints of milk. The next cow gives 5 pints of milk. The next gives 6 pints, and so on, increasing by 1 pint each cow.

pints of milk is increasing 1 by a sequence

4,5,6,7 ......... and so on

when n=1 , the cow given 4 pints of milk

The sequence is arithmetic

nth cow is 199. Lets find out pints of milk when n=199

apply nth formula of arithmetic

[tex]T_n = T_1 + (n-1) d\\[/tex]

n=199

difference d=1

T1=4

Substitute all the values

[tex]T_n = T_1 + (n-1) d\\\\\\T_{199}=4+(199-1)1\\T_{199}=4+198\\T_{199}=202[/tex]

So, the last cow gives 202 pints of milk

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Using the empirical rule, approximately 84% of apple diameters are greater than 7.07 cm, since this value is within one standard deviation below the mean diameter of 7.46 cm.

The empirical rule (also known as the 68-95-99.7 rule) is a statistical rule which states that for a normally distributed set of data, approximately 68% of data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

In this particular problem, we know the mean (7.46cm) and standard deviation (0.39cm) of the apple diameters. We are asked to find the percentage of apple diameters that are greater than 7.07cm. Since 7.07cm is less than one standard deviation below the mean (7.46 - 0.39 = 7.07), from the normal distribution we can say that approximately 84% (50% of the data is above the mean and 34% is between the mean and one standard deviation below the mean) of the apple diameters are larger than 7.07cm according to the Empirical rule.

Thus, using Empirical rule and given statistical mean and standard deviation, we can estimate that about 84% of the apples from this species have a diameter greater than 7.07 cm.

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Using the Empirical Rule, approximately 84% of the apples have diameters greater than 7.07cm since 7.07cm is exactly one standard deviation below the mean of 7.46cm in a normal distribution.

To answer the student's question, we need to apply the Empirical Rule which is used in statistics to describe the distribution of data in a bell-shaped curve, also known as a normal distribution. This rule states that for a data set with a normal distribution, approximately 68% of the data falls within one standard deviation from the mean, 95% falls within two standard deviations, and over 99% falls within three standard deviations.

Given that the mean diameter of the apples is 7.46cm and the standard deviation is 0.39cm, first we need to determine how many standard deviations 7.07cm is from the mean. This is calculated as:

(7.07cm - 7.46cm) / 0.39cm = -1 standard deviation.

According to the Empirical Rule, 68% of apples are within one standard deviation above and below the mean. Since 7.07cm falls exactly at one standard deviation below the mean, we have:

Therefore, to find the percentage of apples that have diameters greater than 7.07cm, we would add these two percentages together:

50% + 34% = 84%

Thus, we conclude that approximately 84% of the apples have diameters that are greater than 7.07cm.