How to find the radius of a sphere - Math

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Question

The volume of a sphere is one cubic yard. Give its radius in inches.

Answer

The volume of a sphere with radius is

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

To find the radius in yards, we set and solve for .

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

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

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

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

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

$r = \sqrt[3]{ $\frac{3}{4\pi}$ } = \sqrt[3]{ $\frac{3\cdot 2 $\pi^{2}$$}{4\pi \cdot 2 $\pi^{2}$} }= \sqrt[3]{ $\frac{6 $\pi^{2}$$}{8 $\pi^{3}$} } = $\frac{ \sqrt[3]{6 $\pi^{2}$$} } {2\pi}$ yards.

Since the problem requests the radius in inches, multiply by 36:

$36 \times $\frac{ \sqrt[3]{6 $\pi^{2}$$} } {2\pi} = $\frac{ 18\sqrt[3]{6 $\pi^{2}$$} } {\pi}$

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Question

If the volume of a sphere is $120\ $ft^{3}$$ , what is the approximate length of its diameter?

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

Answer

The correct answer is 6.12 ft.

Plug the value of into the equation so that

$120=\frac{4}{3}$\pi $r^{3}$$

Multiply both sides by 3 to get

$360=4\pi $r^{3}$$

Then divide both sides by $4\pi$ to get

Then take the 3rd root of both sides to get 3.06 ft for the radius. Finally, you have to multiply by 2 on both sides to get the diameter. Thus

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Question

The volume of a sphere is . What is its radius?

Answer

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

The only given information in the problem is the sphere's final volume. If the volume is , the formula for volume can be used to calculate the sphere's radius.

In this case, , the radius, is the only unknown variable that needs to be solved for.

$$\frac{4}{3}$ \pi $r^3$ = 57$

$\frac{3}{4}$\cdot $\frac{1}{\pi}$\cdot$\frac{4}{3}$ \pi $r^3$ = 57 \cdot$\frac{3}{4}$\cdot $\frac{1}{\pi}$

$r=\sqrt[3]{13.608}$

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Question

The area of a sphere is . What is its radius?

Answer

The only information given is the area of .

This problem may be approached "backwards," where the area formula for a sphere can be used to solve for the radius. This is possible because the formula for area is $Area = 4 \cdot \pi \cdot $r^2$$ , where (the radius) is what we're looking for. After is substituted in for the area, the goal is to solve for by getting it by itself on one side of the equals sign.

$87=4 \cdot \pi \cdot $r^2$$

$\frac{87}{4\cdot\pi}$=\frac{4 \cdot \pi \cdot $r^2$}{4\cdot\pi}$

$$\sqrt{\frac{87}{4 \cdot \pi}$} = $$\sqrt{r^2$}$ = r$

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Question

If the volume of a sphere is , what is the sphere's exact radius?

Answer

Write the formula for the volume of a sphere:

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

Plug in the given volume and solve for the radius, .

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

Start by multiplying each side of the equation by $\frac{3}{4}$ :

$\frac{3}{4}$\cdot1=\frac{4}{3}$\pi $r^3$\cdot$\frac{3}{4}$

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

Now, divide each side of the equation by $\pi$ :

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

Finally, take the cubed root of each side of the equation:

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

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Question

Given the volume of a sphere is , what is the radius?

Answer

The equation for the volume of a sphere is:

$V=\frac{4}{3}$\pi $r^3$$ , where is the length of the sphere's radius.

Plug in the given volume and solve for to calculate the sphere's radius:

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

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Question

If the volume of a sphere is , what is the radius of the sphere?

Answer

The formula for the volume of a sphere is:

$V=\frac{4}{3}$\pi $r^3$$ , where is the sphere's radius.

Plug in the volume and solve for , the sphere's radius:

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

$3=\pi $r^3$$

$r=\sqrt[3]{$\frac{3}{\pi}$}:m$

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Question

Find the radius of a sphere if the surface area is .

Answer

The formula for the surface area of a sphere is:

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

Substitute the given value for the sphere's surface area into the equation and solve for to find the radius:

$8\pi=4\pi $r^2$$

$r=\sqrt2:cm$

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Question

Find the radius of a sphere if its surface area is .

Answer

The surface area formula for a sphere is:

$A=4\pi $r^2$$ , where is the sphere's radius.

Substitute the given value for the sphere's area into the equation and solve for to find the radius:

$8=4\pi $r^2$$

$\frac{2}{\pi}$= $r^2$

$r=$\sqrt{\frac{2}{\pi}$}:in$

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Question

Find the radius of a sphere whose surface area is .

Answer

We know that the surface area of the spere is .

$SA=4\pi $r^{2}$$

Rearrange and solve for .

$\dpi{100} $r=$\sqrt{\frac{100cm^{2}$$}{4\pi}}$

$\dpi{100}\rightarrow 2.82cm$

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Question

What is the radius of a sphere that has a surface area of $16\pi$ ?

Answer

The standard equation to find the area of a sphere is $SA=4\pi $r^2$$ where denotes the radius. Rearrange this equation in terms of :

$r=$\sqrt{\frac{SA}{4\pi }$}$

To find the answer, substitute the given surface area into this equation and solve for the radius:

$r=$\sqrt{\frac{16\pi }{4\pi }$}=$\sqrt{4}$=2$

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Question

Given that the volume of a sphere is $12\pi$ , what is the radius?

Answer

The standard equation to find the volume of a sphere is

$V=\frac{4}{3}$\pi $r^3$$ where denotes the radius. Rearrange this equation in terms of :

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

Substitute the given volume into this equation and solve for the radius:

$r=\sqrt[3]{$\frac{3V}{4\pi }$}=\sqrt[3]{$\frac{3\cdot 12\pi }{4\pi }$}=\sqrt[3]{$\frac{36\pi }{4\pi }$}=\sqrt[3]{9 }$

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Question

What is the radius of a sphere with a volume of $288\pi$ ?

Answer

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

$288\pi=\frac{4}{3}$\pi $r^3$$

$\frac{288\pi}{\pi}$=\frac{4\pi $r^3$}{3\pi}$

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Question

The volume of a sphere is one cubic yard. Give its radius in inches.

Answer

The volume of a sphere with radius is

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

To find the radius in yards, we set and solve for .

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

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

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

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

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

$r = \sqrt[3]{ $\frac{3}{4\pi}$ } = \sqrt[3]{ $\frac{3\cdot 2 $\pi^{2}$$}{4\pi \cdot 2 $\pi^{2}$} }= \sqrt[3]{ $\frac{6 $\pi^{2}$$}{8 $\pi^{3}$} } = $\frac{ \sqrt[3]{6 $\pi^{2}$$} } {2\pi}$ yards.

Since the problem requests the radius in inches, multiply by 36:

$36 \times $\frac{ \sqrt[3]{6 $\pi^{2}$$} } {2\pi} = $\frac{ 18\sqrt[3]{6 $\pi^{2}$$} } {\pi}$

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Question

If the volume of a sphere is $120\ $ft^{3}$$ , what is the approximate length of its diameter?

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

Answer

The correct answer is 6.12 ft.

Plug the value of into the equation so that

$120=\frac{4}{3}$\pi $r^{3}$$

Multiply both sides by 3 to get

$360=4\pi $r^{3}$$

Then divide both sides by $4\pi$ to get

Then take the 3rd root of both sides to get 3.06 ft for the radius. Finally, you have to multiply by 2 on both sides to get the diameter. Thus

Compare your answer with the correct one above

Question

The volume of a sphere is . What is its radius?

Answer

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

The only given information in the problem is the sphere's final volume. If the volume is , the formula for volume can be used to calculate the sphere's radius.

In this case, , the radius, is the only unknown variable that needs to be solved for.

$$\frac{4}{3}$ \pi $r^3$ = 57$

$\frac{3}{4}$\cdot $\frac{1}{\pi}$\cdot$\frac{4}{3}$ \pi $r^3$ = 57 \cdot$\frac{3}{4}$\cdot $\frac{1}{\pi}$

$r=\sqrt[3]{13.608}$

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Question

The area of a sphere is . What is its radius?

Answer

The only information given is the area of .

This problem may be approached "backwards," where the area formula for a sphere can be used to solve for the radius. This is possible because the formula for area is $Area = 4 \cdot \pi \cdot $r^2$$ , where (the radius) is what we're looking for. After is substituted in for the area, the goal is to solve for by getting it by itself on one side of the equals sign.

$87=4 \cdot \pi \cdot $r^2$$

$\frac{87}{4\cdot\pi}$=\frac{4 \cdot \pi \cdot $r^2$}{4\cdot\pi}$

$$\sqrt{\frac{87}{4 \cdot \pi}$} = $$\sqrt{r^2$}$ = r$

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Question

If the volume of a sphere is , what is the sphere's exact radius?

Answer

Write the formula for the volume of a sphere:

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

Plug in the given volume and solve for the radius, .

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

Start by multiplying each side of the equation by $\frac{3}{4}$ :

$\frac{3}{4}$\cdot1=\frac{4}{3}$\pi $r^3$\cdot$\frac{3}{4}$

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

Now, divide each side of the equation by $\pi$ :

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

Finally, take the cubed root of each side of the equation:

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

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Question

Given the volume of a sphere is , what is the radius?

Answer

The equation for the volume of a sphere is:

$V=\frac{4}{3}$\pi $r^3$$ , where is the length of the sphere's radius.

Plug in the given volume and solve for to calculate the sphere's radius:

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

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Question

If the volume of a sphere is , what is the radius of the sphere?

Answer

The formula for the volume of a sphere is:

$V=\frac{4}{3}$\pi $r^3$$ , where is the sphere's radius.

Plug in the volume and solve for , the sphere's radius:

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

$3=\pi $r^3$$

$r=\sqrt[3]{$\frac{3}{\pi}$}:m$

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