How to find the volume of a sphere - High School 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

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 solid hemisphere has a radius of length r. Let S be the number of square units, in terms of r, of the hemisphere's surface area. Let V be the number of cubic units, in terms of r, of the hemisphere's volume. What is the ratio of S to V?

Answer

First, let's find the surface area of the hemisphere. Because the hemisphere is basically a full sphere cut in half, we need to find half of the surface area of a full sphere. However, because the hemisphere also has a circular base, we must then add the area of the base.

$S = \frac{1}{2}\cdot$ (surface area of sphere) + (surface area of base)

The surface area of a sphere with radius r is equal to $4\pi r^2$ . The surface area of the base is just equal to the surface area of a circle, which is $\pi r^2$ .

$S=\frac{1}{2}\cdot 4\pi r^2+\pi r^2=2\pi r^2+\pi r^2=3\pi r^2$

The volume of the hemisphere is going to be half of the volume of an entire sphere. The volume for a full sphere is $\frac{4}{3}\pi r^3$ .

$V = \frac{1}{2}\cdot$ (volume of sphere)

$V = \frac{1}{2}\cdot \frac{4}{3}\pi r^3=\frac{2}{3}\pi r^3$

Ultimately, the question asks us to find the ratio of S to V. To do this, we can write S to V as a fraction.

$\frac{S}{V}=\frac{3\pi r^2}{\frac{2}{3}\pi r^3}$

In order to simplify this, let's multiply the numerator and denominator both by 3.

$\frac{S}{V}=\frac{3\pi r^2}{\frac{2}{3}\pi r^3}$ = $\frac{9\pi r^2}{2\pi r^3}=\frac{9}{2r}$

The answer is $\frac{9}{2r}$ .

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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 radius of ?

Answer

To solve for the volume of a sphere, you must first know the equation for the volume of a sphere.

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

In this equation, is equal to the radius. We can plug the given radius from the question into the equation for .

$V=\frac{4}{3}(\pi)(12^{3})$

Now we simply solve for .

$V=\frac{4}{3}(\pi)(1728)$

$V=(\pi)(2304)=2304\pi$

The volume of the sphere is $2304\pi$ .

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Question

What is the volume of a sphere with a radius of 4? (Round to the nearest tenth)

Answer

To solve for the volume of a sphere you must first know the equation for the volume of a sphere.

The equation is

Then plug the radius into the equation for yielding

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

Then cube the radius to get

$V=\frac{4}{3}(64)\pi$

Multiply the answer by $\frac{4}{3}$ and to yield $85.3\pi$ .

The answer is $85.3\pi$ .

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Question

A typical baseball is in diameter. Find the baseball's volume in cubic centimeters.

Answer

In order to find the volume of a sphere, use the formula

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

We were given the baseball's diameter, $\dpi{100} D=76mm$ , which must be converted to its radius.

$r=\frac{76mm}{2}$

$\rightarrow 38mm$

Now we can solve for volume.

$V=\frac{4}{3}\pi (38mm)^{3}$

$V=\frac{4}{3}\pi (54872mm^{3})$

$V=73162\frac{2}{3}mm^{3}\pi$

$\rightarrow 229847.30mm^{3}$

Convert to centimeters.

$\dpi{100} \frac{229847.30mm^{3}}{1}\frac{1cm^{3}}{1000mm^{3}}\approx 230cm^{3}$

If you arrived at $1838cm^{3}$ then you did not convert the diameter to a radius.

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Question

What is the volume of a sphere whose radius is .

Answer

In order to find the volume of a sphere, use the formula

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

We were given the radius of the sphere, .Therefore, we can solve for volume.

$\dpi{100} V=\frac{4}{3}\pi (1.6in)^{3}$

$\dpi{100} V=\frac{4}{3}\pi (4.096in^{3})$

$\dpi{100} V=17.16in^{3}$

If you calculated the volume to be $\dpi{100} 9.65in^{3}$ then you multiplied by $\frac{3}{4}$ rather than by $\frac{4}{3}$ .

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Question

To the nearest tenth of a cubic centimeter, give the volume of a sphere with surface area 1,000 square centimeters.

Answer

The surface area of a sphere in terms of its radius is

$A = 4\pi r^2$

Substitute and solve for :

$4\pi r^2 =A$

$4\pi r^2 =1,000$

$r^2 =\frac{1,000}{4\pi } = \frac{250}{\pi }$

$r =\sqrt{ \frac{250}{\pi }} \approx 8.92 \textrm{ cm }$

Substitute for in the formula for the volume of a sphere:

$V = \frac{4\pi r^3}{3} \approx\frac{4\pi \cdot 8.92^3}{3} \approx 2,973.5 \textrm{ cm }^{3}$

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Question

Find the volume of the following sphere.

Answer

The formula for the volume of a sphere is:

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

where is the radius of the sphere.

Plugging in our values, we get:

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

$V=\frac{4}{3} \pi 216m^3 = 288 \pi m^3$

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Question

Find the volume of the following sphere.

Answer

The formula for the volume of a sphere is:

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

Where is the radius of the sphere

Plugging in our values, we get:

$V = \frac{4 \pi (9m)^3}{3}$

$V = 972 \pi m^3$

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Question

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

Answer

The formula for volume of a sphere is $V=\frac{4}{3}\pi r^3$ .

The problem gives us the diameter, however, and not the radius. Since the diameter is twice the radius, or , we can find the radius.

$\frac{12}{2}=r$

.

Now plug that into our initial equation.

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

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

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

$V=288\pi$

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