This little pool is 10 feet in diameter and 3 feet tall. Use the formula for volume to find the total water needed to fill the pool.

V=π r ^2 h

Explanation:

The Law of Sines is your friend, as is the Pythagorean theorem.

Label the unmarked slanted segments "a" and "b" with "b" being the hypotenuse of the right triangle, and "a" being the common segment between the 45° and 60° angles.

Then we have from the Pythagorean theorem ...

  b² = 4² +(2√2)² = 24

  b = √24

From the Law of Sines, we know that ...

  b/sin(60°) = a/sin(θ)

  y/sin(45°) = a/sin(φ)

Solving the first of these equations for "a" and the second for "y", we get ...

  a = b·sin(θ)/sin(60°)

and ...

  y = a·sin(45°)/sin(φ)

Substituting for "a" into the second equation, we get ...

  y = b·sin(θ)/sin(60°)·sin(45°)/sin(φ) = (b·sin(45°)/sin(60°))·sin(θ)/sin(φ)

So, we need to find the value of the coefficient ...

  b·sin(45°)/sin(60°) = (√24·(√2)/2)/((√3)/2)

  = √(24·2/3) = √16 = 4

and that completes the development:

  y = 4·sin(θ)/sin(φ)

This does NOT represent a function.

There should only be one pair per input and output, but this graph has multiple.

Answer:

94

Step-by-step explanation:

Jen has

Answer: The height of the tree is 12.57 ft.

Step-by-step explanation:

The height of a thing is proportional to the its shadow.

Let H = Height and S = length of shadow

Then by equation direct proportion , [tex]\dfrac{H_1}{S_1}=\dfrac{H_2}{S_2}[/tex]       (i)

Given : A tree that casts a 22 ft shadow at the same time a 4 ft postpost casts a shadow which is 7-ft ​long.

Put [tex]S_1=22 ,\ H_2= 4,\ \ S_2=7[/tex]  in (i), we get

[tex]\dfrac{H_1}{22}=\dfrac{4}{7}\\\\ H_1=\dfrac{4}{7}\times22\approx12.57[/tex]

Hence, the height of the tree is 12.57 ft.

What the user said above!

Answer:

57

Step-by-step explanation:

(f•g)(x) = f(x)•g(x)

= (4x+3)(2x-5)

= 8x²-20x+6x-15

= 8x²-14x-15

(f•g)(4) = [4(4)+3] [2(4)-5]

= (16+3)(8-5)

= 19 • 3

= 57

Answer:

57

Step-by-step explanation:

got it correct on test. :)

Answer:

Total magnification of microscope at this setting is 40X

Step-by-step explanation:

Total magnification of microscope is determined by multiplying the magnification power of individual lenses.

So if eyepiece has magnification power of 10X and objective lense has magnification power of 4X , the total magnification of microscope would be

10 × 4 = 40

which means the object will appear 40 times larger than actual object.

The total magnification of the microscope at this setting is 40X.

To determine the total magnification of a compound microscope, one needs to multiply the magnification power of the eyepiece by the magnification power of the objective lens. In this case, the eyepiece has a magnification power of 10X and the objective lens has a magnification power of 4X.

The formula for the total magnification [tex]\( M_{total} \)[/tex] is given by:

[tex]\[ M_{total} = M_{eyepiece} \times M_{objective} \][/tex]

Substituting the given values:

[tex]\[ M_{total} = 10X \times 4X \][/tex]

[tex]\[ M_{total} = 40X \][/tex]

Therefore, the total magnification of the microscope when using the 10X eyepiece and the 4X objective lens is 40X. This means that the image seen through the microscope will appear 40 times larger than its actual size.

Answer: the average weight of the remaining 2 non-blue items is 81 pounds

Step-by-step explanation:

The formula for determining average is expressed as

Average = sum of each item/ total number of items

Let x represent the average weight of the remaining 2 non-blue items.

The average weight of 5 items is 24 pounds. If the total weight of 3 blue items is 39 pounds, it means that

(x + 39)/5 = 24

x + 39 = 5 × 24 = 120

x = 120 - 39

x = 81

Answer:

δhsoln will be close to zero.

Step-by-step explanation:

In calculus, the symbol d represents a large or significant increment in a value. For example, say the change in the volume of a liquid in a tank depends upon the change of the height h.

The above statement can be written like this:

dV/dh

This means that the volume of the tank (V) depends with a significant change in the height of the liquid (h).

It is also possible to compute small changes in physical quantities. The symbol δ simply presents a small increment or small change. Using the same expression above, if a very large tank was to have a very very small leak, the change would be: δV/δh

In other words, the change in the volume will be almost negligible and will be close to zero.  

Answer:

So, 22 batches of cookies can the baker make.

Step-by-step explanation:

A baker has 15 cups of flour.

He sets aside 4 cups.

So, the remaining cups = 15 - 4 = 11

A cookie recipe that calls for 1/2 cup of flour.

So, x batches of cookies can the baker make with 11 cups

Use the ratio and proportional

1 : 1/2 = x : 11   ⇒ multiply both sides by 2

2 : 1 = 2x : 22

solve for x

2x = 2 * 22

x = 22

So, 22 batches of cookies can the baker make.

She can make 22 batches of cookies.

Given that there are 15 cups of flour. The baker sets aside 4 cups.

So remaining (15 - 4) = 11 cups.

Let she can make x batches.

Given each batch takes 1/2 cup.

That means she will need [tex]x\cdot\frac{1}{2}=\frac{x}{2}[/tex] cups.

According to the question:

[tex]\frac{x}{2}=11\\x=22[/tex]

So she can make 22 batches of cookies.

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Answer:

Time = 9.0252 *10^12 s

Step-by-step explanation:

Given:

- The distance from Earth to Alpha Centauri = 4.3 light years

- The distance from Earth to Sirius = 8.6 light years

- Speed of the probe is V = 18.03 km/s

- 1 AU = 1.58125 x 10-5 light-years

Find:

How long will the probe have been traveling from when it first leaves Earth to when it arrives at Sirius?

Solution:

- We will track probe for each destination it reaches one by one:

                       Earth ------> Alpha Centauri d_1 = 4.3 light years

                       Alpha Centauri ------> Earth d_2 =4.3 light years

                       Earth ------> Sirius                  d_3 = 8.6 light years  

                       Total distance                       D    = 17.2 light years.

- Now we know the total distance traveled by the probe is D. We will convert the distance into km SI units:

                          1 AU    ------------------> 1.58125 x 10-5 light-years

                          x AU   ------------------> 17.2 light years.

- Using direct proportions

                           x = 17.2 / (1.58125 x 10-5) = 1087747.036 AU

Also,

                          1 AU  ---------------------> 149597870700 m

                           1087747.036 AU  ---->    D m

- Using direct proportions

                            D =   1087747.036*149597870700  = 1.62725*10^17 m

- Now use the speed - distance - time formula to compute the total time taken:

                            Time = Distance traveled (D) / V

                            Time = 1.62725*10^17 / 18.03*10^3    

Answer:               Time = 9.0252 *10^12 s

Answer:142

2

+2

Step-by-step explanation:

Answer:

6 meters

Step-by-step explanation:

Intersecting Chord Theorem: When two chords intersect each other inside a circle, the products of their segments are equal.

One chord is divided into two segments with lengths of 15 m and 2 m.

Anothe chord is divided into two segments with measures of 5 m and n m.

Therefore,

[tex]5\cdot n=15\cdot 2\\ \\5n=30\\ \\n=6\ m[/tex]

Answer:

6 m

Step-by-step explanation:

just took the test

Answer: the cost of one apple is $1

Step-by-step explanation:

Let x represent the cost of one apple.

Let y represent the cost of one Pomegranate.

Donata bought 3 Apples and 5 Pomegranates at the local supermarket for a total of $16.50. This means that

3x + 5y = 16.5 - - - - - - - - - - - - -1

Meaghan bought 6 Apples and 11 Pomegranates at the same store for a total of $35.70. This means that

6x + 11y = 35.7- - - - - - - - - - - - -2

Multiplying equation 1 by 2 and equation 2 by 1, it becomes

6x + 10y = 33

6x + 11y = 35.7

Subtracting, it becomes

- y = - 2.7

y = 2.7

Substituting y = 2.7 into equation 1, it becomes

3x + 5 × 2.7 = 16.5

3x + 13.5 = 16.5

3x = 16.5 - 13.5 = 3

x = 3/3 = 1

Answer:

Area = 823 * 534 = 439,482 ft^2

Step-by-step explanation:

Answer:

[tex]50w+100 \ge 18000[/tex]

Step-by-step explanation:

Each seven-letter word is worth 50 points. [tex]w[/tex] words are then worth [tex]w\times50=50w[/tex].

Since Sal gets 100 points for playing the game, his points total after [tex]w[/tex] words is [tex]50w +100[/tex]. If he wants to break the record, he must get, at least, 18000 points. "At least" in inequality means [tex]\ge[/tex] (just as "at most" means [tex]\le[/tex]). Then the required inequality is

[tex]50w+100 \ge 18000[/tex].

Note that the question says "18000 or more", which is why we used the [tex]\ge[/tex] symbol. If the phrase had been "more than 18000", we would have used [tex]>[/tex].

Answer:

6

Step-by-step explanation:

8^2*(TS)^2=10^2

(TS)^2=36

TS=6

Answer:

Step-by-step explanation:

Triangle RST is a right angle triangle.

From the given right angle triangle,

RS represents the hypotenuse of the right angle triangle.

With m∠R as the reference angle,

RT represents the adjacent side of the right angle triangle.

ST represents the opposite side of the right angle triangle.

To determine m∠R, we would apply

the cosine trigonometric ratio.

Cos θ = adjacent side/hypotenuse. Therefore,

Cos m∠R = 8/10 = 0.8

m∠R = Cos^-1(0.8)

m∠R = 36.9 to one decimal place.

Answer:

The correct answer for this equational ratio is x = 5

Step-by-step explanation:

In order to calculate for x, we must create the necessary ratios

[tex]\frac{16}{20} = \frac{45-5x}{5x}[/tex]

Cross multiply to get

900 - 100x = 80x

900 = 180x

x = 5

Answer:

C. 2

Step-by-step explanation:

Given:

[tex]\frac{5}{12}x-\frac12=\frac13[/tex]

We need to solve the equation to find the value of x.

Solution:

[tex]\frac{5}{12}x-\frac12=\frac13[/tex]

Combining like terms we get;

[tex]\frac{5}{12}x=\frac13+\frac12[/tex]

Now Using LCM for making the denominators same we get;

[tex]\frac{5}{12}x=\frac{1\times2}{3\times2}+\frac{1\times3}{2\times3}\\\\\frac{5}{12x}=\frac{2}{6}+\frac{3}{6}[/tex]

Now Denominators are common so we solve the number we get;

[tex]\frac{5}{12}x=\frac{2+3}{6}\\\\\frac{5}{12}x=\frac{5}{6}[/tex]

Now Dividing both side by [tex]\frac{12}{5}[/tex] we get;

[tex]\frac{5}{12}x\times \frac{12}{5}=\frac{5}{6}\times \frac{12}{5}\\\\x=2[/tex]

Hence the value of x is 2.

Answer:

C. f(x) = 2^x+2 - 3

Step-by-step explanation:

Looking at the graph and answer choices, it's obvious that the only change is the vertical shift. Since we know that the smallest value of 2^x+2 is 0, we can infer that the vertical shift will be -3 to match the horizontal asymptote of y=-3.

After 10 s, the bucket has rotated [tex]180^{\circ}[/tex]

Step-by-step explanation:

The motion of the bucket on the water wheel is a periodic motion, which means that it repeats itself periodically. The period of the motion is the time it takes for the bucket to returns to its original position: it can be found from the graph by looking at after how much time the graph has again the same shape.

We see that if we  start from x = 0 seconds, then the graph has the same shape again (=it has completed one full cycle) at x = 20 seconds, so the period is

T = 20 s

This also means that in a time of 20 seconds, the bucket has covered a full revolution, so an angle of

[tex]\theta=360^{\circ}[/tex]

Therefore, in order to find how many degrees x the bucket has rotated in

t = 10 s

We  can use the rule of three:

[tex]\frac{\theta}{T}=\frac{x}{t}[/tex]

And solving for x, we find

[tex]x=\frac{\theta t}{T}=\frac{(360)(10)}{20}=180^{\circ}[/tex]

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Answer:

  (n -4)(n +1)^2 = n^3 -2n^2 -7n -4

Step-by-step explanation:

The least common denominator is the smallest denominator that lets you write the sum as a single fraction.

  [tex]\dfrac{n^5}{n^2+2n+1}+\dfrac{-4}{n^2-3n-4}=\dfrac{n^5}{(n+1)^2}+\dfrac{-4}{(n+1)(n-4)}\\\\=\dfrac{n^5}{(n+1)^2}\cdot\dfrac{n-4}{n-4}+\dfrac{-4}{(n+1)(n-4)}\cdot\dfrac{n+1}{n+1}=\dfrac{n^5(n-4)-4(n+1)}{(n+1)^2(n-4)}\\\\=\dfrac{n^6-4n^5-4n-4}{n^3-2n^2-7n-4}[/tex]

The least common denominator is ...

  (n-4)(n+1)^2 = n^3 -2n^2 -7n -4

Jim's first error occurred when he incorrectly applied the cube to three-fifths. He made the mistake of not applying the power to the denominator of the fraction. The answer is  54/125

The question revolves around a mathematical operation where Jim is attempting to determine the cube of 2 times three-fifths. Jim's first error lies in how he applied the power of three. According to the rules of exponentiation for fractions, you should apply the power to both the numerator and denominator. Hence, Jim's first mistake was that he did not apply the power to the denominator of three-fifths. Instead of merely cubing the numerator (3), Jim should have also cubed the denominator (5). He should have performed the calculation as follows:

= 2 x (3/5)^3

= 2 x (3^3/5^3)

= 2 (27/125).

= 54/125

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Jim first errored by failing to apply the exponent to both the numerator and denominator of the fraction before multiplying by two. The correct calculation for cubing a fraction should involve cubing both the numerator and the denominator, which would have resulted in 54/125 instead of 54/5.

The first error that Jim made was not applying the power to both the numerator and the denominator of the fraction before multiplying by two. When cubing a number in fractional form, such as three-fifths, one must cube both the numerator (33) and the denominator (53) to correctly apply the exponent to the entire fraction. The correct method is to first cube three-fifths, which results in 27/125, and then multiply that result by 2, leading to the final answer of 54/125, not 54/5 as Jim calculated.

It's important to follow the correct order of operations and apply exponents before multiplication. In this case, to find the value of 2 × (three-fifths) cubed, the calculation should be 2 × (3/5)3, which simplifies to 2 × (27/125), and then to 54/125.

Answer:

9.6

Step-by-step explanation:

The question is :

Solution;

Let total distance from school to home = 'x' meters

Issa jogged 2/3rd of 'x'= [tex]\frac{2x}{3}[/tex]

So total distance from school to home = [tex]\frac{2x}{3} +3200[/tex]

But total distance from school to home = [tex]x[/tex]

[tex]==> \frac{2x}{3} + 3200 = x[/tex]

[tex]==> 3200= \frac{x}{3}[/tex]

[tex]==> 9600=x[/tex]

So total distance from school to home is 9600 meters

We convert it into km now

we know 1000 meters = 1km

==> 9600 meters = 1/100 x 9600= 9.6 km

Answer:

9.6 or (48\5

Step-by-step explanation:

Answer:

Step-by-step explanation:

This is an exponential function:

[tex]y=16(\frac{2}{3})^x[/tex] that tells us that the initial height of the ball is 16 feet and that after each successive bounce the ball comes up to 2/3 its previous height.  We are looking for y when x = 2, so

[tex]y=16(\frac{2}{3})^2[/tex] and

[tex]y=16(\frac{4}{9})[/tex] and

[tex]y=\frac{64}{9}[/tex] so

y = 7.1 feet

Answer:

a) (5,0)

b) (2,0)

Step-by-step explanation:

(a) A particle that lies 5.0 m directly above the origin would have its x-coordinate be 5 and its y-coordinate be 0. So (5,0).

(b) A particle that lies 2.0 m directly below the origin would have its x-coordinate be 2 and its y-coordinate be 0. So (2,0).

In a coordinate system with the positive x-axis directed upward, a particle located 5.0 m directly above the origin has a position of +5.0 m, and a particle 2.0 m below the origin has a position of -2.0 m.

In this coordinate system, the position of a particle is indicated by its vertical position relative to the origin. Therefore:

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Answer:

h = 6 units.

Step-by-step explanation:

Given:

Area of the parallelogram = [tex]54\ units^2[/tex].

Base = 9 units.

We need to find the missing height.

Solution:

We know that the area of the parallelogram is represented by below formula.

[tex]A = bh[/tex] -----------(1)

Where:

A = Area of the parallelogram.

h = height of the parallelogram.

b = Base

Since, base and area is known. So, we substitute these values in equation 1.

[tex]54 = 9\times h[/tex]

[tex]h = \frac{54}{9}[/tex]

h = 6 units.

Therefore, height of the parallelogram h = 6 units.

Answer:

98 degrees

Step-by-step explanation:

m<H = 45, m<F = 53

180 = 45 + 53 + (180-x)

82 = 180-x

x+82 = 180

x = 98

Answer:

Step-by-step explanation:

The sum of the angles in a triangle is 180 degrees. This means that

Angle F + angle G + angle H = 180

Therefore,

53 + 45 + angle G = 180

98 + angle G = 180

Subtracting 98 from the left hand side and the right hand side of the equation, it becomes

98 - 98 + angle G = 180 - 98

angle G = 82 degrees

The sum of angles on a straight line is 180 degrees. Therefore,

x + 82 = 180

Subtracting 82 from the left hand side and the right hand side of the equation, it becomes

x + 82 - 82 = 180 - 82

x = 98 degrees

Answer:

The required inequality is [tex]0.0001 x^2 - 0.089 x - 7<0[/tex].

Step-by-step explanation:

The given inequalities are

[tex]A(x) = 0.0051x^2 - 0.319x + 15[/tex]

[tex]V(x)= 0.005x^2 - 0.23x + 22[/tex]

where, x is the driver's age (in years), A(x) is driver’s reaction time to audio stimuli and V(x) is his or her reaction time to visual stimuli, 16 ≤ x ≤ 70.

We need to find an inequality that can be use to find the x-values for which A(x) is less than V(x).

[tex]A(x)<V(x)[/tex]

[tex]0.0051x^2 - 0.319x + 15< 0.005x^2 - 0.23x + 22[/tex]

[tex]0.0051x^2 - 0.319x + 15- 0.005x^2 + 0.23x- 22<0[/tex]

Combine like terms.

[tex]0.0001 x^2 - 0.089 x - 7<0[/tex]

where, 16 ≤ x ≤ 70.

Therefore, the required inequality is [tex]0.0001 x^2 - 0.089 x - 7<0[/tex].

There are 85 candy bars in each box, and he will have 340 candy bars left after he gives three boxes to a friend.

Answer: he has 340 candy bars left.

Step-by-step explanation:

The total number of candy bars that

Miguel ordered is 595.

They come in 7 boxes. Assuming each box contains equal number of candy bars. This means that the number of candy bars in each box would be

595/7 = 85 candy bars

If he gives 3 boxes of candy bars to his friend, it means that the number of candy bars that he gave to his friend is

85 × 3 = 255 candy bars

Therefore, the number of candy bars that he has left is

595 - 255 = 340

Answer:

Side length = [tex]\sqrt{\frac{A}{3} }[/tex] cm ,   Height =  [tex]\frac{1}{2} \sqrt{\frac{A}{3} }[/tex] cm  ,  Volume = [tex]\frac{A\sqrt{A}}{6\sqrt{3} }[/tex]  cm³

Step-by-step explanation:

Assume

Side length of base = x

Height of box = y

total material required to construct box = A ( given in question)

So it can be written as

A = x² + 4xy

4xy = A - x²

Volume of box = Area x height

V = x² ₓ y

V = x² ₓ ( [tex]\frac{A - x^{2} }{4x}[/tex] )

V =  [tex]\frac{Ax - x^{3} }{4}[/tex]

To find max volume put V' = 0

So taking derivative equation becomes

[tex]\frac{A - 3 x^{2} }{4} = 0[/tex]

A = 3 [tex]x^{2}[/tex]

[tex]x^{2}[/tex] = [tex]\frac{A}{3}[/tex]

x = [tex]\sqrt{\frac{A}{3\\} }[/tex]

put value of x in equation 1

y = [tex]\frac{A - \frac{A}{3} }{4\sqrt{\frac{A}{3} } }[/tex]  

y = [tex]\frac{2 \sqrt{\frac{A}{3} } }{4 \sqrt{\frac{A}{3} } }[/tex]

y = [tex]\frac{1}{2} \sqrt{\frac{A}{3} }[/tex]

So the volume will be

V = [tex]x^{2}[/tex] × y

Put values of x and y from equation 2 & 3

V = [tex]\frac{A}{3} (\frac{1}{2} \sqrt{\frac{A}{3} } )[/tex]

V = [tex]\frac{A\sqrt{A}}{6\sqrt{3} }[/tex]

The side length is [tex]\mathbf{l =\sqrt{ \frac A3}}[/tex], the height is [tex]\mathbf{h = \frac{1}{2}\sqrt{\frac A3}}[/tex] and the maximal volume is  [tex]\mathbf{V = \frac A6 \sqrt{\frac A3}}[/tex]

Let the dimensions of the box be l and h, where l represents the base length and h represents the height.

The volume is calculated as:

[tex]\mathbf{V = l^2h}[/tex]

The surface area is:

[tex]\mathbf{A= l^2 + 4lh}[/tex]

Make h the subject

[tex]\mathbf{h = \frac{A- l^2}{4l}}[/tex]

Substitute [tex]\mathbf{h = \frac{A- l^2}{4l}}[/tex] in [tex]\mathbf{V = l^2h}[/tex]

[tex]\mathbf{V = l^2 \times \frac{A- l^2}{4l}}[/tex]

[tex]\mathbf{V = l \times \frac{A- l^2}{4}}[/tex]

[tex]\mathbf{V = \frac{Al- l^3}{4}}[/tex]

Split

[tex]\mathbf{V = \frac{Al}{4}- \frac{l^3}{4}}[/tex]

Differentiate

[tex]\mathbf{V' = \frac{A}{4}- \frac{3l^2}{4}}[/tex]

Set to 0

[tex]\mathbf{\frac{A}{4}- \frac{3l^2}{4} = 0}[/tex]

Multiply through by 4

[tex]\mathbf{A- 3l^2 = 0}[/tex]

Add 3l^2 to both sides

[tex]\mathbf{3l^2 = A}[/tex]

Divide both sides by 3

[tex]\mathbf{l^2 = \frac A3}[/tex]

Take square roots

[tex]\mathbf{l =\sqrt{ \frac A3}}[/tex]

Recall that: [tex]\mathbf{h = \frac{A- l^2}{4l}}[/tex]

So, we have:

[tex]\mathbf{h = \frac{A - \frac{A}{3}}{4\sqrt{A/3}}}[/tex]

[tex]\mathbf{h = \frac{\frac{2A}{3}}{4\sqrt{A/3}}}[/tex]

Divide

[tex]\mathbf{h = \frac{2\sqrt{A/3}}{4}}[/tex]

[tex]\mathbf{h = \frac{\sqrt{A/3}}{2}}[/tex]

Rewrite as:

[tex]\mathbf{h = \frac{1}{2}\sqrt{\frac A3}}[/tex]

Recall that:

[tex]\mathbf{V = l^2h}[/tex]

So, we have:

[tex]\mathbf{V = \frac A3 \times \frac{1}{2}\sqrt{\frac A3}}[/tex]

[tex]\mathbf{V = \frac A6 \sqrt{\frac A3}}[/tex]

Hence, the side length is [tex]\mathbf{l =\sqrt{ \frac A3}}[/tex], the height is [tex]\mathbf{h = \frac{1}{2}\sqrt{\frac A3}}[/tex] and the maximal volume is  [tex]\mathbf{V = \frac A6 \sqrt{\frac A3}}[/tex]

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