Spheres - ACT Math

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Question

The specifications of an official NBA basketball are that it must be 29.5 inches in circumference and weigh 22 ounces. What is the approximate volume of the basketball? Remember that the volume of a sphere is calculated by V=(4πr3)/3

Answer

To find your answer, we would use the formula: C=2πr. We are given that C = 29.5. Thus we can plug in to get \[29.5\]=2πr and then multiply 2π to get 29.5=(6.28)r. Lastly, we divide both sides by 6.28 to get 4.70=r. Then we would plug into the formula for volume V=(4π〖(4.7)〗3) / 3 (The information given of 22 ounces is useless)

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Question

Find the volume of a sphere whose diameter is .

Answer

To solve, simply use the formula for the volume of a sphere. Thus,

$V=\frac{4}{3}\pi{r^3}=\frac{4}{3}\pi2^3=\frac{4}{3}8\pi=\frac{32}{3}\pi$

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Question

If the diameter of a sphere is , find the approximate volume of the sphere?

Answer

The volume of a sphere = $\frac{4}{3}\pi r^{3}$

Radius is $\frac{1}{2}$ of the diameter so the radius = 5.

$V=\frac{4}{3}\pi5^{3}$

or $V=\frac{500}{3\pi}$

which is approximately $167\pi$

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Question

For a sphere the volume is given by V = (4/3)πr_3 and the surface area is given by A = 4_πr_2. If the sphere has a surface area of 256_π, what is the volume?

Answer

Given the surface area, we can solve for the radius and then solve for the volume.

4_πr_2 = 256_π_

4_r_2 = 256

_r_2 = 64

r = 8

Now solve the volume equation, substituting for r:

V = (4/3)π(8)3

V = (4/3)π*512

V = (2048/3)π

V = 683_π_

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Question

A cube has a side dimension of 4. A sphere has a radius of 3. What is the volume of the two combined, if the cube is balanced on top of the sphere?

Answer

$V_{cube}=(\text{side})^3=(4)^3=64$

$V{sphere}=\frac{4}{3}\pi r^3=\frac{4}{3}\pi(3^3)=\frac{4}{3}\pi(27)=36\pi$

$V_{total}=64+36\pi$

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Question

What is the volume of a sphere with a diameter of 6 in?

Answer

The formula for the volume of a sphere is:

$V=\frac{4}{3}\pi r^{3}$

where = radius. The diameter is 6 in, so the radius will be 3 in.

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Question

The radius of a sphere is . What is the approximate volume of this sphere?

Answer

$Volume=\frac{4}{3}\pi r^{3}$

$V=\frac{4}{3}\pi(6)^3$

$V=\frac{4}{3}\pi(216)$

$V=288\pi$

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Question

What is the volume of a sphere with a diameter of inches? Leave your answer in terms of $\pi$ .

Answer

To find the volume of a sphere we use the sphere volume formula:

$V = \frac{4}{3}\pi r^3$

First we need to find the radius of the sphere. A sphere has a radius of half the diameter. So we see that .

Next we plug 6 in for our radius and get
$V = \frac{4}{3}\pi 6^3=288\pi in^3$

(don't forget your units).

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Question

What is the volume of a sphere with a diameter of $\textup{6 inches}$ (reduce all fractions)?

Answer

The formula for the volume of a sphere is:
$v = \frac{4}{3}\pi * r^3$ , thus we need to just determine the radius and plug it into the equation. Remember that and so

$6 = 2r \newline 3 = r$

And plugging in we get $\frac{108}{3}\pi \textup{in}^3$

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Question

What is the volume of a sphere with a surface area of $576\pi\textup{ ft}^2$ ? (Simplify all fractions in your answer.)

Answer

First find the radius from the surface area set the given surface area equal to the surface area formula and solve for the radius.
$576\pi = 4\pi r^2$

$144\pi = \pi r^2 \newline 144 = r^2 \newline 12 = r$

Now plug the radius into the volume formula:
$V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi 12^3 = 2304 \pi\textup{ ft}^3$

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Question

If Ariana’s orange has twice the radius of Autumn’s orange, the volume of Ariana’s orange is how many times larger than the volume of Autumn’s orange?

Answer

Define the radius of Autumn’s orange as r. The volume of her orange is $\pi r^3$ . Ariana’s orange has twice the radius of Autumn’s, so the radius of her orange is , and the volume is $\pi(2r)^3 = 8\pi r^3$ , which is 8 times larger than Autumn’s orange.

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Question

If a sphere has a volume of $36\pi$ , what is its diameter?

Answer

1. Use the volume to find the radius:

$Volume=\frac{4}{3}\pi r^{3}$

$36\pi = \frac{4}{3}\pi r^{3}$

$\frac{3}{4}(36\pi) = (\frac{4}{3}\pi r^{3})\frac{3}{4}$

$27=r^{3}$

2. Use the radius to find the diameter:

$d=2r=2\cdot 3=6$

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Question

A sphere has a volume of $36\pi$ . What is its diameter?

Answer

This question relies on knowledge of the formula for volume of a sphere, which is as follows: $V=\frac{4}{3}\pi r^{3}}$

In this equation, we have two variables, and . Additionally, we know that $V=36\pi$ and is unknown. You can begin by rearranging the volume equation so it is solved for , then plug in and solve for :

Rearranged form:

$r=\sqrt[3]{\frac{3}{4\pi}V}$

Plug in $36\pi$ for V

$r=\sqrt[3]{\frac{3}{4\pi}36\pi}$

Simplify the part under the cubed root

Cancel the $\pi$ 's since they are in the numerator and denominator.

Simplify the fraction and the :

$36*\frac{3}{4}=27$

Thus we are left with

$r=\sqrt[3]{27}$

Then, either use your calculator and enter $27^{\frac{1}{3}}$ Or recall that $3^{3}=27$ in order to find that .

We're almost there, but we need to go a step further. Dodge the trap answer "" and carry on. Read the question carefully to see that we need the diameter, not the radius.

So

is our final answer.

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Question

A spherical plastic ball has a diameter of . What is the volume of the ball to the nearest cubic inch?

Answer

To answer this question, we must calculate the volume of the ball using the equation for the volume of a sphere. The equation for the volume of a sphere is four-thirds multiplied by pi, which is then multiplied by the radius cubed. The equation can be written like this:

$V = \frac{4}{3}\pi r^{3}$

We are given the diameter of the sphere in the problem, which is . To get the radius from the diameter, we divide the diameter by . So, for this data:

$radius=\frac{diameter}{2}=\frac{4}{2}=2$

We can then plug our newly found radius of two into the equation to find the volume. For this data:

$Volume = \frac{4}{3}\pi r^{3}=\frac{4}{3}\pi \cdot (2)^{3} = \frac{4}{3}\pi\cdot 8$

We then multiply $\frac{4}{3}$ by .

$\frac{4}{3}\pi\cdot 8=\frac{32}{3}\pi$

We finally substitute 3.14 for pi and multiply again to get our answer.

$\frac{32}{3}\pi = 33.5$

The question asked us to round to the nearest whole cubic inch. To do this, we round a number up one place if the last digit is a 5, 6, 7, 8, or 9, and we round it down if the last digit is a 1, 2, 3, or 4. Therefore:

$33.5\rightarrow34$

Therefore our answer is .

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Question

A boulder breaks free on a slope and rolls downhill. It rolls for complete revolutions before grinding to a halt. If the boulder has a volume of cubic feet, how far in feet did the boulder roll? (Assume the boulder doesn't lose mass to friction). Round $\pi$ to 3 significant digits. Round your final answer to the nearest integer.

Answer

The formula for the volume of a sphere is:

$V = \frac{4}{3}\pi r^3$

To figure out how far the sphere rolled, we need to know the circumference, so we must first figure out radius. Solve the formula for volume in terms of radius:

$1436 = \frac{4}{3}\pi r^3$

$\frac{1077}{\pi} =r^3$

$r \approx 6.999932$

Since the answer asks us to round to the nearest integer, we are safe to round to at this point.

To find circumference, we now apply our circumference formula:

$C = 2r\pi = 14\pi$

If our boulder rolled times, it covered that many times its own circumference.

$355\cdot 14\pi = 4970\pi \approx 15606$

Thus, our boulder rolled for

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Question

Find the diameter of a sphere whose radius is .

Answer

To solve, simply remember that diameter is twice the radius. Don't be fooled when the radius is an algebraic expression and incorporates the arbitrary constant . Thus,

$\textup{diameter}=2r=2*2d=4d$

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Question

The surface area of a sphere is $64\pi$ feet. What is the radius?

Answer

Solve the equaiton for the surface area of a sphere for the radius and plug in the values:

$SA=4{\pi}r^2\rightarrow r=\sqrt{\frac{SA}{4\pi}}= \sqrt{\frac{64\pi}{4\pi}}=\sqrt16=4$

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Question

What is the radius of a sphere with a volume of $2304\pi$ ? Round to the nearest hundredth.

Answer

Recall that the equation for the volume of a sphere is:

$V=\frac{4}{3}\pi r^3$

For our data, we know:

$2304\pi=\frac{4}{3}\pi r^3$

Solve for . First, multiply both sides by $\frac{3}{4}$ :

$\frac{3}{4}2304\pi=\pi r^3$

$1728\pi=\pi r^3$

Now, divide out the $\pi$ :

Using your calculator, you can solve for . Remember, if need be, you can raise to the power of $\frac{1}{3}$ if your calculator does not have a variable-root button.

This gives you:

If you get something like , just round up. This is a rounding issue with some calculators.

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Question

The volume of a sphere is $972\pi:cm^3$ . What is the diameter of the sphere? Round to the nearest hundredth.

Answer

Recall that the equation for the volume of a sphere is:

$V=\frac{4}{3}\pi r^3$

For our data, we know:

$972\pi = \frac{4}{3}\pi r^3$

Solve for . Begin by dividing out the $\pi$ from both sides:

$972 = \frac{4}{3} r^3$

Next, multiply both sides by $\frac{3}{4}$ :

$\frac{3}{4} * 972 =r^3$

Using your calculator, solve for . Recall that you can always use the $\frac{1}{3}$ power if you don't have a variable-root button.

You should get:

If you get , just round up to . This is a general rounding problem with calculators. Since you are looking for the diameter, you must double this to .

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Question

What is the radius of a sphere with a surface area of $169\pi$ ? Round to the nearest hundredth.

Answer

Recall that the surface area of a sphere is found by the equation:

$SA = 4\pi r^2$

For our data, this means:

$169\pi = 4\pi r^2$

Solve for . First, divide by $4\pi$ :

Take the square root of both sides:

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