Aaron and Blake both wrote down equation. Their equation has the same slope, But Aaron's y-intercept was negative and Blake's y-intercept was positive. How many solutions for their system of equations have? Explain your reasoning

Answer:

75% increase

Step-by-step explanation:

Given:

Number of missed days of work before the management change = 4

Number of missed days of work after the management change = 7

So, increase in the number of missed days of work is given by subtracting the old number from the new number and is equal to:

Increase in missed number of days = 7 - 4 = 3 days

Now, the percent of this increase is calculated by dividing the increase by the old number and then multiplying the result by 100.

Percent of increase is given as:

[tex]Percent\ of\ increase=\frac{Increased\ value}{Old\ value}\times 100\\\\Percent\ of\ increase=\frac{3}{4}\times 100\\\\Percent\ of\ increase=0.75\times 100\\\\Percent\ of\ increase=75\%[/tex]

So, there is a 75% increase in the number of missed days of work after the change in the management.

Answer:

Therefore,

[tex]x=9\\\\y=7[/tex]

Step-by-step explanation:

Given:

D,E,F are the midpoints of side AB, BC,and AC such that

AB = 4x - 18

BC = 2x - 4

DF = x

FE = y

To Find:

x = ?

y = ?

Solution:

Midpoint Theorem:

The Midpoint Theorem states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side.  

E and F are Midpoint of Sides BC and AC, then

[tex]FE=\dfrac{1}{2}AB[/tex]        ......By Midpoint Theorem

Substituting the values we get

[tex]x=\dfrac{1}{2}(4x-18)=2x-9\\\\2x-x=9\\\\x=9[/tex]

Similarly,

D and Fare Midpoint of Sides AB and AC, then

[tex]DF=\dfrac{1}{2}BC[/tex]        ......By Midpoint Theorem

Substituting the values we get

[tex]y=\dfrac{1}{2}(2x-4)=x-2\\\\Substitute\ x=9\\\\y=9-2=7[/tex]

Therefore,

[tex]x=9\\\\y=7[/tex]

Answer:

Step-by-step explanation:

We would apply the formula for exponential decay which is expressed as

A = P(1 - r/n)^ nt

Where

A represents the value after t years.

n represents the period for which the decrease in value is calculated

t represents the number of years.

P represents the value population.

r represents rate of decrease.

From the information given,

P = 23000

r = 8% = 8/100 = 0.08

n = 1

Therefore, the exponential decay function described in this situation is

A = 23000(1 - 0.08/n)1)^ 1 × t

A = 23000(0.92)^t

If A = 15000, then

15000 = 23000(0.92)^t

0.92^t = 15000/23000 = 0.6522

Taking log of both sides to base 10

Log 0.92^t = log 0.6522

tlog 0.92 = log 0.6522

- 0.036t = - 0.1856

t = - 0.1856/- 0.036

t = 5 years to the nearest year

The value of the car will be $15,000 after approximately 4 years.

To model the situation with an exponential decay function, we use the formula:

[tex]\[ V(t) = P \times (1 - r)^t \][/tex]

Given that the initial cost [tex]P[/tex] is $23,000 and the annual depreciation rate r is  8%, or 0.08 in decimal form, we can write the exponential decay function as:

[tex]\[ V(t) = 23000 \times (1 - 0.08)^t \][/tex]

[tex]\[ V(t) = 23000 \times (0.92)^t \][/tex]

To find out when the value of the car will be $15,000, we set [tex]V(t)[/tex] equal to $15,000 and solve for t.

[tex]\[ \frac{15000}{23000} = (0.92)^t \][/tex]

[tex]\[ 0.652173913043478 = (0.92)^t \][/tex]

Using a calculator, we find:

[tex]\[ t \approx \frac{\ln(0.652173913043478)}{\ln(0.92)} \approx 4.037 \][/tex]

Since we are looking for the time in whole years, we round to the nearest year:

[tex]\[ t \approx 4 \][/tex]

 Therefore, the value of the car will be $15,000 after approximately 4 years.

Answer: 6 minutes

Step-by-step explanation:

According to the Empirical rule ,

About 68% of the population lies within one standard deviation from the mean.

Given : Suppose the wait through immigration at JFK Airport in New York is thought to be bell-shaped and symmetrical with a mean of 22 minutes.

It is known that 68 percent of travelers will spend between 16 and 28 minutes waiting to pass through immigration.

i.e. Mean - standard deviation= 16

and Mean  + standard deviation =28

Since , Mean=22 , so we have

22+ standard deviation =28

⇒ standard deviation =28 -22 =6

Hence, the standard deviation for the wait time through immigration is 6 minutes .

The standard deviation for the wait time through immigration is 6 minutes.

To find the standard deviation, we can use the properties of the normal distribution, specifically the 68-95-99.7 rule, which states that approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

Given that 68% of travellers wait between 16 and 28 minutes, we can determine that these times represent one standard deviation from the mean. The mean wait time is 22 minutes. The range from 16 to 28 minutes is 12 minutes wide (28 - 16 = 12). Since this range represents one standard deviation on either side of the mean, we can divide this range by 2 to find the value of one standard deviation:

Standard deviation = (28 - 16) / 2 = 12 / 2 = 6 minutes.

Anna bought 10 oranges

Explanation:

Let x denote the number of apples.

Let y denote the number of oranges.

The system of equations can be written as

[tex]x+y=25[/tex] and [tex]2x+3y=60[/tex]

Let us solve the equation by substitution method.

Thus, from [tex]x+y=25[/tex], the value of y can be determined such that [tex]y=25-x[/tex]

Using the y value [tex]y=25-x[/tex] in the equation [tex]2x+3y=60[/tex], we get,

[tex]2x+3(25-x)=60[/tex]

Multiplying 3 within the bracket,

[tex]2x+75-3x=60[/tex]

Adding the terms,

[tex]75-x=60[/tex]

Subtracting both sides by 75, we have,

[tex]x=15[/tex]

Thus, substituting [tex]x=15[/tex] in [tex]y=25-x[/tex], we get,

[tex]y=25-15\\y=10[/tex]

Thus, Anna get 10 Oranges.

Answer:

The (minimum) number of toothpicks required to make an acute angle scalene triangle is  3.

Step-by-step explanation:

The (minimum) number of toothpicks required to make an acute angle scalene triangle is  3.

All the angles in a an acute angled scalene triangle are less than 90°.

The above condition can be satisfied with a minimum of three toothpicks.

To create an acute-angled scalene triangle, three toothpicks are required.

To create an acute-angled scalene triangle, you would need 3 toothpicks. You can stick the toothpicks into a marshmallow or a gumdrop and position them in a way that forms a triangle with angles less than 90 degrees and with all sides of different lengths.

Answer:$4000 was invested into the account earning 6% interest.

$2000 was invested into the account earning 11% interest.

Step-by-step explanation:

Let x represent the amount invested into the account earning 6% interest.

Let y represent the amount invested into the account earning 11% interest.

Mike has $6000 to invest. He invested part of his total into an account earning 6% interest and the rest in an account earning 11% interest. This means that

x + y = 6000

The formula for simple interest is expressed as

I = PRT/100

Where

P represents the principal

R represents interest rate

T represents time in years

I = interest after t years

Considering the account earning 6% interest, the interest would be

I = (x × 6 × 1)/100 = 0.06x

Considering the account earning 11% interest, the interest would be

I = (x × 11 × 1)/100 = 0.11y

If at the end of the year, he earned $460.00 in interest, it means that

0.06x + 0.11y = 460 - - - - - - - - - - -1

Substituting x = 6000 - y into equation 1, it becomes

0.06(6000 - y) + 0.11y = 460

360 - 0.06y + 0.11y = 460

- 0.06y + 0.11y = 460 - 360

0.05y = 100

y = 100/0.05 = $2000

x = 6000 - y = 6000 - 2000

x = $4000

Answer:

Therefore the period of the sinusoidal wave = 3.14.

The amplitude of the function = 2.

Step-by-step explanation:

The period of a sinusiodal wave is given by the length of x axis which covers one full cycle of the wave, which one positive half-cycle and one negative half cycle.

From the graph we can see  that one of the cycles starts at -3.14 / 4 and ends at 3.14 [tex]\times[/tex] 3/4.

Therefore we can sutract the start point value from the end point value to get the period

Therefore period  = (3.14 [tex]\times[/tex] 3/4) - (-3.14 / 4) = (3.14 [tex]\times[/tex] 3/4) + (3.14 / 4) = 3.14

Therefore the period of the sinusoidal wave = 3.14.

The amplitude of the function = 2.

Answer:

  tan(23° -21°) = (tan(23°) -tan(21°))/(1 +tan(23°)tan(21°))

Step-by-step explanation:

The formula for the tangent of the difference of angles is ...

  tan(a-b) = (tan(a) -tan(b))/(1 +tan(a)tan(b))

Filling in the values a=23° and b=21°, you get the formula shown above.

The expression for tan(23°−21°) in terms of the tangents of 23° and 21° is: tan(23°−21°) = (tan 23° - tan 21°) / (1 + tan 23° * tan 21°).

To express tan(23°−21°) in terms of the tangents of 23° and 21°, we can use the angle subtraction formula for tangent, which is:

tan(α - β) = (tan α - tan β) / (1 + tan α * tan β)

Applying this to tan(23°−21°), we get:

tan(23°−21°) = (tan 23° - tan 21°) / (1 + tan 23° * tan 21°)

We can't directly calculate the values from the given table since 23° and 21° are not listed. However, we understand the formula required to express tan(23°−21°) in terms of the tangents of these two angles.

Answer:

175 means the city's population is 175 thousands in 2010.

Step-by-step explanation:

The model below gives the projection of the city's population , P, in thousands, with respect to time, t, in years, where 2010 corresponds to t=0.

The given equation is

[tex]P=175+\dfrac{11}{2}t[/tex]

We need to interpret 175 for the given model.

Substitute t=0 in the above equation.

[tex]P=175+\dfrac{11}{2}(0)[/tex]

[tex]P=175[/tex]

At t=0 the value of P is 175 it means the city's population is 175 thousands in 2010.

Answer:

For a scaler variable, the Gaussian distribution has a probability density function of

p(x |µ, σ² ) = N(x; µ, σ² ) = 1 / 2π×[tex]e^{\frac{-(x-u)^{2}}{2s^{2} } }[/tex]

The term will have a maximum value at the top of the slope of the 1-D Gaussian distribution curve that is when exp(0) =1 or when x = µ

Step-by-step explanation:

Gaussian distributions have similar shape, with the mean controlling the location and the variance controls the dispersion  

From the graph of the probability distribution function it is seen that the the peak is the point at which the slope = 0, where µ = 0 and σ² = 1 then solution for the peak = exponential function = 0 or x = µ

Answer:

A'(8, -10)

Step-by-step explanation:

Firstly we have to know the distance from the point A to the line, the point is (8, 2) and the line is x = 5 .

the point has two components, at x and y

the component in y will give us the distance we have on the axis of the X thas is 2.

T he line is 5 units from the axis.

Now to know the distance we have to subtract

2 - 5 = -3

Here is the point with respect to the line x = 5

For the reflected over the line we just need to change the sign, now we got +3 .  To this we add the value of the line that is 5

3 + 5 = 8.

So the reflected point over the line x = 5 is (8, 8)

We do the same with the line x = -1.

(8, 8) and x = -1

..

8 - (-1) = 9

..

(-9) + (-1) = -10

(8, -10)

Answer:

At most 29 boxes can be safely loaded onto the pallet.

Step-by-step explanation:

Given:

Weight of each box = 55 pound

Weight pallet can support [tex]\leq[/tex] 1595 pounds.

We need to find the number of boxes .

Solution:

Let the number of boxes loaded safely onto the pallet be 'x'.

So we can say that;

Weight of each box Multiplied by number of boxes should be less than or equal to Weight pallet can support.

framing in equation form we get;

[tex]55x\leq 1595[/tex]

Dividing both side by 55 we get;

[tex]\frac{55x}{55}\leq \frac{1595}{55}\\\\x\leq 29[/tex]

Hence At most 29 boxes can be safely loaded onto the pallet.

Alan can safely load 29 boxes onto the pallet.

To determine how many boxes Alan can safely load onto the pallet, we need to divide the maximum weight the pallet can support by the weight of each box. The maximum weight the pallet can support is 1595 pounds, and each box weighs 55 pounds.

To find the number of boxes, we can divide the maximum weight by the weight of each box: 1595 ÷ 55 = 29.

Therefore, Alan can safely load 29 boxes onto the pallet.

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Answer:

[tex]\frac{3}{8}[/tex]  of a box of candy your friend eat.

Step-by-step explanation:

Given:

You and your friend are going to some candy.

You eat 3/4 of a box of candy.

Your friend eats 1/2 as much as you do.

Now, to find that how much of a box of candy does your friend eat.

Quantity of a box of candy you eat = [tex]\frac{3}{4}[/tex]

Quantity of candy your friend eat = [tex]\frac{1}{2} \ of\ \frac{3}{4}[/tex]

Now, to get how much the candy your friend eat we multiply 1/2 and 3/4:

= [tex]\frac{1}{2} \times \frac{3}{4}[/tex]

= [tex]\frac{3}{8}[/tex] .

Therefore, [tex]\frac{3}{8}[/tex]  of a box of candy your friend eat.

Answer:

  neither

Step-by-step explanation:

The equation means that 600 is twelve times as many as 50 (not four times). Bill needs to rethink the meaning of "four" relative to the meaning of "12".

__

"650 more than 12" is an expression (12+650), not an equation. Sarah seems to be clueless.

__

Neither Bill nor Sarah have described the meaning of the equation.

Answer: The amount she puts in a bag  = 0.912 pound.

Step-by-step explanation:

Given : Amount of red licorice purchased by Christy = 6.75 pounds

Amount of black licorice purchased by Christy  = 2.37 pounds

Total amount of licorice purchased by Christy =  6.75 + 2.37 pounds

= 9.12  pounds

Then, if she divides the total amount into 10 equal-sized bags  , the amount she put in a bag  = (Total amount of licorice purchased) ÷ 10

= 9.12  ÷ 10 = 0.912 pound

Hence, the amount she puts in a bag  = 0.912 pound.

Final answer:

To find how much licorice Christy needs to put into each of 10 bags, add the weights of red and black licorice and then divide by 10. She will need to put 0.912 pounds of licorice into each bag.

Explanation:

Christy purchased 6.75 pounds of red licorice and 2.37 pounds of black licorice. To find the total amount of licorice, we add the weight of the red and black licorice together:

Total weight of licorice = [tex]6.75 pounds + 2.37 pounds = 9.12 pounds.[/tex]

If she divides the total amount into 10 equal-sized bags, we divide the total weight by 10:

Weight per bag = [tex]9.12 pounds / 10 bags = 0.912 pounds per bag.[/tex]

Answer: $135.42

Step-by-step explanation:

Based on IRS tables, Mary is expected to receive 192 (16 years x 12 months) annuity payments. Her investment in the annuity is $70,000 and her return of capital for each annuity payment is $70,000/192 = $364.58. The return of capital portion of each annuity payment is not taxable (not included in gross income). Mary must include the excess received ($500.00 – 364.58) of $135.42 in her gross income.

To calculate how much of the first $500 payment Mary will include in her gross income, we need to determine the exclusion ratio. The exclusion ratio is the ratio of the investment in the annuity to the expected return. In this case, Mary will include $36.45 of the first $500 payment in her gross income.

To determine how much of the first $500 payment Mary will include in her gross income, we need to calculate the exclusion ratio. The exclusion ratio is the ratio of the investment in the annuity to the expected return. In this case, Mary paid $70,000 for the annuity and her life expectancy is 16 years, so the exclusion ratio is $70,000 / ($500/month * 12 months/year * 16 years) = 0.0729. To determine the amount she will include in her gross income, we multiply the monthly payment by the exclusion ratio: $500 * 0.0729 = $36.45. Therefore, Mary will include $36.45 of the first $500 payment in her gross income.

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Answer:

Step-by-step explanation:

Given that  a school has 20 classes: 16 with 25 students in each, three with 100 students in each, and one with 300 students for a total of 1000 students.

a) Y = no of students in the class out of 20 selected at random

Y can take values as

Y          25     100    300   Total

freq      16        3          1       20

P(Y)     16/20   3/20   1/20     1

y*p      20        15       15       50

E(Y) = 50

b) X = size of student from which one student is taken at random

X                                    25       100    300     Total

No of classes                16           3         1         20

Total students          

(X*no of classes)           400    300    300     1000

P(x)                                 0.4      0.3       0.3          1

X*P                                 160       90      90       240

E(x) =240

c)   On an average,   a student at this school is in a class with 240 students. On average, a class at this school has 50 students.

i.e. expectation of number of students in a class is 50 while expectation of a student having students in the class is 240  

Answer:

x = 144 children

y = 160 adults

Step-by-step explanation:

We have parameters already defined. Number of children that entered is x while number of admitted adults is y.

Since the total number of people admitted was 304, this means:

x + y = 304

Total fees for that day was 856

Mathematically:

1.5x + 4y = 856

From the first equation, we can say y = 304 - x

We then substitute this into the second equation:

1.5x + 4(304 - x) = 856

1.5x + 1216 - 4x = 856

1216-856 = 4x-1.5x

2.5x = 360

x = 360/2.5 = 144

Since y = 304 - 144

y = 160

The number of children admitted was 144 and the number of adult is 160

Answer: the value of the account after 10 years is $2606

Step-by-step explanation:

The formula for continuously compounded interest is

A = P x e (r x t)

Where

A represents the future value of the investment after t years.

P represents the present value or initial amount invested

r represents the interest rate

t represents the time in years for which the investment was made.

e is the mathematical constant approximated as 2.7183.

From the information given,

P = 1800

r = 3.7% = 3.7/100 = 0.037

t = 10 years

Therefore,

A = 1800 x 2.7183^(0.037 x 10)

A = 1800 x 2.7183^(0.37)

A = $2606 to the nearest dollar

To solve such problems we must know about Continuous Compound Interest.

[tex]A = Pe^{rt}[/tex]

where,

A is the principal amount after t number of years,

r is the rate at which the principal is been compounded, and P is the principal amount.

The value of Steve's account after 10 years will be $2,606.

As it is given in the problem, the account of Steve is compounding continuously.

Steve invests 1,800 in an account that earns 3.7% annual interest, compounded continuously.

Time period money invested for 10 years,

Substituting the values in the formula of Continuous Compound Interest,

[tex]A = Pe^{rt}[/tex]

[tex]\begin{aligned}A&=\$1800\times e^{0.037\times10}\\ &=\$2605.9\approx\$2606\\ \end{aligned}[/tex]

Hence, the value of Steve's account after 10 years will be $2,606.

Learn more about Compound Interest:

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Answer:

Lenght = 20m

Width = 30m

Step-by-step explanation:

Perimeter of a rectangular enclosure = 2(L+W)

Adding a length of the fence parallel to one side of the sudden to split the closure, the total is 2(L+W) + L = 120

Area of a rectangular closure = L*W

To find the dimension that would maximize the area solve for L and W.

2(L+W) + L = 120

2L + 2W + L = 120

3L + 2W = 120

2W = 120 - 3L

W = (120 - 3L)/2

Put the value of W into Area formula

A = L*W

A = L *(120 - 3L) /2

= L(60 - 1.5L)

= 60L - 1.5L^2

= -1.5L^2 + 60L

This is a quadratic equation. Compare to ax^2 + bx + c

L = -b/2a

a = -1.5, b= 60, c= 0

L = -60/2(-1.5)

= -60/-3

= 20m

Put L = 20 into the value of W

W = (120 - 3L) /2

W= (120 - 3*20) / 2

= (120 - 60)/2

= 60 / 2

= 30m

The dimensions are 20m by 30m

Answer:

The outcome of guessing the correct answers to a 50 question test is 12.5.

Step-by-step explanation:

Let us first consider one question.

Since it has 4 options so we can say that;

Total number of possible outcomes = 4

Since 1 answer is correct so we can say that;

Favorable outcome = 1

Now we can say that;

Probability for choosing correct answer can be calculated by Favorable outcome divided by Total number of possible outcomes.

framing in equation form we get;

[tex]P(Correct\ Answer) = \frac{1}{4}[/tex]

Now Given:

Total number of question = 50

So we can say that;

Total number of correct answer out of 50 is equal to Probability for choosing correct answer multiplied by Total number of question.

framing in equation form we get;

Guessing number of correct answer out of 50 = [tex]\frac14 \times 50 = 12.5[/tex]

Hence The outcome of guessing the correct answers to a 50 question test is 12.5.

You can use a binomial distribution model to simulate guessing the correct answers on a 50-question test with 4 answer choices per question.  The probability of getting at least 30 correct answers.

To simulate the outcome of guessing the correct answers to a 50 question test, you can use a binomial distribution model. A binomial distribution represents the probability of a certain number of successful outcomes in a fixed number of independent trials. In this case, the success would be guessing the correct answer.

Each question has 4 answer choices, so the probability of guessing the correct answer for each question is 1/4. The number of successful guesses in a 50-question test can follow a binomial distribution with parameters n = 50 (number of trials) and p = 1/4 (probability of success).

Using this model, you can calculate probabilities of various outcomes, such as the probability of getting exactly 20 correct answers by guessing

Hence, the probability of getting at least 30 correct answers.

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Final answer:

To calculate the number of floors a building can have for a given height, use the formula 'Number of Floors = h / 12', where 'h' is the total building height in feet and each floor is 12 feet tall.

Explanation:

To write an expression for the number of floors a building can have for a given building height, where each floor is 12 feet tall, we use the following formula:

Number of Floors = Total Building Height (in feet) / Height of One Floor (in feet)

Let 'h' represent the total height of the building in feet. Then the expression becomes:

Number of Floors = h / 12

For example, if a building has a height of 384 feet, the number of floors can be calculated as 384 feet / 12 feet per floor = 32 floors.

Answer:

Step-by-step explanation:

The value of the y-intercept is (0,7)

So the y intercept is 0,7

Answer:

The sampling method described in this question appears to be sound

Step-by-step explanation:

It appears to be sound because the data are not biased in any way, since every member and set of members has an equal chance of being included in the sample selection. Also, random samples are usually fairly representative since they don't favor certain members.

Answer:

True

Step-by-step explanation:

A common denominator of the mixed numbers is 20. select True or false

Mr. Clements painted his barn for 3 3/5 hours in the morning.

[tex]3\frac{3}{5}=\frac{18}{5}[/tex]

He painted the barn for 5 3/4 hours in the afternoon.

[tex]5\frac{3}{4}=\frac{23}{4}[/tex]

Now we consider the denominator of both fractions 18/5 and 23/4

Least common denominator is 5  times 4 = 20

So its true that the common denominator of mixed numbers is 20

Answer: the distance from the base of the ladder from the building is 10 feet.

Step-by-step explanation:

The ladder forms a right angle triangle with the wall of the building and the ground.

The length of the ladder represents the hypotenuse of the right angle triangle.

The distance, h of the base of the ladder from the building represents the adjacent side of the triangle.

To determine h, we would apply

the cosine trigonometric ratio.

Cos θ = adjacent side/hypotenuse. Therefore,

Cos 60 = h/20

h = 20Cos60 = 20 × 0.5

h = 10 feet

Final answer:

To solve for the distance from the base of the ladder to the building, the cosine function is used because the scenario forms a right-angled triangle with the ground and the wall. By the equation cos(60°) = Base / Hypotenuse, with the ladder being the hypotenuse, the base is calculated to be 10 feet from the building.

Explanation:

The question involves using trigonometry to determine the distance from the base of the ladder to the building. When a ladder makes a 60° angle with the ground and its length is given (20 feet), this forms a right-angled triangle with the ground and the building's wall. To find the distance from the base of the ladder to the building, we can use the cosine function, which relates the adjacent side to the hypotenuse in a right-angled triangle.

Using the equation:

cos(60°) = Base / Hypotenuse

Here, the hypotenuse is the length of the ladder which is 20 feet. Therefore:

cos(60°) = Base / 20 feet

cos(60°) equals 0.5 when calculating in degrees. Consequently:

0.5 = Base / 20 feet

Multiplying both sides by 20 gives:

Base = 20 feet * 0.5

Base = 10 feet

Thus, the base of the ladder is 10 feet from the building.

Answer:

3. A matched-pairs t-test with H0 : μd = 0,  HA : μd ≠ 0

Step-by-step explanation:

An investigator wants to asses the difference in mean pulses rates between sitting and standing persons in population.

The matched paired t-test is appropriate choice because the observations in the data are paired. The observations in the data are the calculated pulse rates of 50 persons while sitting and standing. For each person. sitting and standing pulse rates are measured and this indicates the before and after effect is measured. The before and after effect is measured through matched paired t test.

Also,  an investigator wants to determine whether there is difference in mean pulse rates between sitting and standing in population. This indicates two tailed test. so, the hypotheses would be

Null hypotheses : μd = 0

Alternative hypotheses: μd ≠ 0.

Thus, situation indicates a matched-pairs t-test with H0 : μd = 0,  HA : μd ≠ 0.

The option that represents an appropriate test and hypotheses to determine if there is a difference in mean pulse rates between sitting and standing in the population is: Option C: A matched-pairs t-test with [tex]H_0 : \mu_d = 0\\H_A : \mu_d \neq 0[/tex]

The two-sample t-test is used to determine if two population means are equal.

When there has to be done comparison between means of two sets of paired data, then we use matched pair t test.

There are two hypotheses. First one is called null hypothesis and it is chosen such that it predicts nullity or no change in a thing. It is usually the hypothesis against which we do the test. The hypothesis which we put against null hypothesis is alternate hypothesis.

Null hypothesis is the one which researchers try to disprove.

For this case, we want to determine if there is a difference in mean pulse rates between sitting and standing in the population.

That is the reason why we will use matched pair t-test.

Where paired data is the pair of data obtained from each of those 50 people, the pair of data being sitting pulse rate and standing pulse rate.

Now, since we want to determine if there's any difference, so the null hypothesis will nullify the presence of any difference.

Thus, we get:

Null hypothesis = [tex]H_0: \mu_d = 0[/tex] (the difference between mean of sitting pulse rate and that of standing pulse rate is 0, therefore no difference).

This can be rewritten as: [tex]H_0: \mu_{\text{sitting}} = \mu_{\text{standing}}[/tex]

Alternate hypothesis = [tex]H_A: \mu_d \neq 0[/tex] or [tex]H_A: \mu_{\text{sitting}} \neq \mu_{\text{standing}}[/tex]

Thus, the option that represents an appropriate test and hypotheses to determine if there is a difference in mean pulse rates between sitting and standing in the population is: Option C: A matched-pairs t-test with [tex]H_0 : \mu_d = 0\\H_A : \mu_d \neq 0[/tex]

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Answer:

Step-by-step explanation:

                    30/71 = 0.4225 = 42.25%

Answer:

13π/2 m²/h

Step-by-step explanation:

Volume of a cylinder is:

V = πr²h

If h is a constant, then taking derivative of V with respect to time:

dV/dt = 2πrh dr/dt

Surface area of a cylinder is:

A = 2πr² + 2πrh

Taking derivative with respect to time:

dA/dt = (4πr + 2πh) dr/dt

Given that dV/dt = 10π, V = 80π, and h = 5, we need to find dA/dt.  But first, we need to find r and dr/dt.

V = πr²h

80π = πr² (5)

r = 4

dV/dt = 2πrh dr/dt

10π = 2π (4) (5) dr/dt

dr/dt = 1/4

dA/dt = (4πr + 2πh) dr/dt

dA/dt = (4π (4) + 2π (5)) (1/4)

dA/dt = 13π/2

The surface area of the cylinder is increasing at 13π/2 m²/h.

The required surface area of the cylinder is [tex]\frac{13}{2}[/tex] m²/h.

Given that,

The volume of the cylinder increase with the rate of 10π cubic meters per hour.

The height of the cylinder is fixed at 5 meters.

At a certain instant, the volume is 80π cubic meters.

We have to determine,

What is the rate of change of the surface area of the cylinder at that instant.

According to the question,

Height of cylinder = 5m

Volume of cylinder increase with rate =  10π cubic meters per hour.

At certain instant volume become =  80π cubic meters per hour.

Volume of cylinder is given as,

V = πr²h

Where h is a constant,

[tex]v = \pi r^{2} h\\\\80\pi = \pi r^{2}. (5)\\\\\frac{80\pi }{5\pi } = r^{2} \\\\r^{2} = 16\\\\r = 4[/tex]

Then,  taking derivative of V with respect to time:

[tex]\frac{dv}{dt} = 2\pi rh.\frac{dr}{dt} \\\\[/tex]

Where, dv\dt = 10π, V = 80π, and h = 5,

Then,

[tex]10\pi = 2\pi (4)(5). \frac{dr}{dt} \\\\10\pi = 40\pi \frac{dr}{dt} \\\\\frac{10\pi }{40\pi } = \frac{dr}{dt} \\\\\\\frac{dr}{dt} = \frac{1}{4}[/tex]

Then,

Surface area of a cylinder is define as,

[tex]A = 2\pi r^{2} + 2\pi rh\\\\\ \frac{da}{dt} = (4\pi r + 2\pi h)\frac{dr}{dt} \\\\\frac{da}{dt} = (4\pi (4) + 2\pi (5))\times\frac{1}{4} \\\\\frac{da}{dt} = \frac{26\pi }{4} \\\\\frac{da}{dt} = \frac{13\pi }{2}[/tex]

Hence, The required surface area of the cylinder is [tex]\frac{13}{2}[/tex] m²/h.

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