How to find the volume of a sphere - Math

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Question

In terms of $\pi$ , give the volume, in cubic feet, of a spherical tank with diameter 36 inches.

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Answer

36 inches = $36 \div 12 = 3$ feet, the diameter of the tank. Half of this, or $\frac{3}{2}$ feet, is the radius. Set $r = $\frac{3}{2}$$ , substitute in the volume formula, and solve for :

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

$V = $\frac{4}{3}$ \pi\left \cdot \left( $\frac{3}{2}$ \right ) ^{3}$

$V = $\frac{4}{3}$\cdot $\frac{3}{2}$ \cdot $\frac{3}{2}$ \cdot $\frac{3}{2}$ \cdot \pi\left$

$V = $\frac{1}{1}$\cdot $\frac{1}{1}$ \cdot $\frac{3}{1}$ \cdot $\frac{3}{2}$ \cdot \pi\left$

$V = $\frac{9}{2}$ \pi$

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Question

Which is the greater quantity?

(a) The volume of a sphere with radius

(b) The volume of a cube with sidelength

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Answer

A sphere with radius has diameter and can be inscribed inside a cube of sidelength . Therefore, the cube in (b) has the greater volume.

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Question

Which is the greater quantity?

(a) The volume of a cube with sidelength inches.

(b) The volume of a sphere with radius inches.

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Answer

You do not need to calculate the volumes of the figures. All you need to do is observe that a sphere with radius inches has diameter inches, and can therefore be inscribed inside the cube with sidelength inches. This give the cube larger volume, making (a) the greater quantity.

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Question

Which is the greater quantity?

(a) The volume of a sphere with diameter one foot

(b) $864 \textrm{ $in}^{3}$$

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Answer

The radius of the sphere is one half of its diameter of one foot, which is six inches, so substitute :

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

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

$V = $\frac{4}{3}$ \pi \cdot 216$

$V = 288 \pi \approx 288 \cdot 3.14 = 904.32$ cubic inches,

which is greater than $864 \textrm{ $in}^{3}$$ .

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Question

is a positive number. Which is the greater quantity?

(A) The volume of a cube with edges of length

(B) The volume of a sphere with radius

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Answer

No calculation is really needed here, as a sphere with radius - and, subsequently, diameter - can be inscribed inside a cube of sidelength . This makes (A), the volume of the cube, the greater.

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Question

Which is the greater quantity?

(a) The radius of a sphere with surface area $36 \pi$

(b) The radius of a sphere with volume $36 \pi$

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Answer

The formula for the surface area of a sphere, given its radius , is

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

The sphere in (a) has surface area $36 \pi$ , so

$36 \pi = 4 \pi $r^{2}$$

$36 \pi\div 4 \pi = 4 \pi $r^{2}$ \div 4 \pi$

$9 = $r^{2}$$

$r = $\sqrt{9}$ = 3$

The formula for the volume of a sphere, given its radius , is

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

The sphere in (b) has volume $36 \pi$ , so

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

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

$27 \pi = \pi $r^{3}$$

$27 \pi\div \pi = \pi $r^{3}$ \div \pi$

$27 = $r^{3}$$

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

The radius of both spheres is 3.

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Question

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

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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

A sphere has a circumference of $8\pi$ , what is its volume?

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Answer

The circumference is given by $2\pi r$ which yields a radius of 4. The volume is given by $\frac{4}{3}$\pi $r^{3}$

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Question

Given a sphere of radius 7cm, what is the volume of the sphere?

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Answer

$\frac{4}{3}$\cdot \pi \cdot $r^3$\rightarrow $\frac{4}{3}$\cdot \pi \cdot $(7)^3$

Simplified, it is $\frac{1372}{3}$; \pi ; $cm^3$

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Question

What is the volume of a sphere with surface area 1,000 square centimeters?

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Answer

Use the surface area formula to find the radius, then use the volume formula to find the volume.

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

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

$r =$\sqrt{\frac{250}{\pi }$}$

$V=\frac{4}{3}$ \pi $r^{3}$ = $\frac{4}{3}$ \pi \left($\sqrt{\frac{250}{\pi }$} \right $)^{3}$ \approx 2,974$

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Question

The circumference of a sphere is $27 \pi :cm$ . What is the sphere's volume?

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Answer

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

Finding the volume is simple. All that we need is the radius!

The only information the problem provides is the circumference. In order to find the radius, we have to think how circumference relates to radius.

Since the equation for circumference is $C=\pi \cdot d$ , where d stands for diameter, and radius is half of the diameter, the two have diameter in common.

The first step in solving this problem is to determine the diameter from the circumference:

$C= \pi \cdot d$

$27 \cdot \pi = \pi \cdot d$

$\frac{27 \cdot \pi}{ \pi}$ = $\frac{\pi \cdot d}{\pi}$

Because the diameter is , the radius must be .

Now we are ready to solve for the volume after substituting in our value.

$V= $\frac{4}{3}$ \cdot \pi \cdot $(13.5)^3$$

$V= $\frac{4}{3}$ \cdot \pi \cdot (2,460.375)$

$V = 3280.5 \cdot \pi$

$V =10,305.99 $\approx10,306:cm^3$$

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Question

A cube with a side length of 20 inches has a sphere inscribed within. What is the volume of the sphere?

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Answer

The cube has a side length of 20 inches. Since the sphere is inscribed within the cube its diameter measures 20 inches; the radius will be 10 inches.

The volume of a sphere is given by

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

$V=\frac{4}{3}$\pi $(10)^3$$=\mathbf{4,189\hspace{1mm}in^3$}$

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Question

A sphere is cut in half as shown by the figure below.

If the radius of the sphere is , find the volume of the figure.

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Answer

Recall how to find the volume of a sphere:

$\text{Volume of $Sphere}$=\frac{4}{3}$\pi\times$\text{radius}$^3$

Now since we only have half a sphere, divide the volume by .

$\text{Volume of $Figure}$=\frac{1}{2}$($\frac{4}{3}$\pi\times$\text{radius}$)^2$

$\text{Volume of $Figure}$=\frac{2}{3}$\pi\times$\text{radius}$^3$

Plug in the given radius to find the volume of the figure.

$$\text{Volume of $Figure}$=\frac{2}{3}$\pi(2)^3$=\frac{16}{3}$\pi=16.76$

Make sure to round to places after the decimal.

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Question

A sphere is cut in half as shown by the figure below.

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

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Answer

Recall how to find the volume of a sphere:

$\text{Volume of $Sphere}$=\frac{4}{3}$\pi\times$\text{radius}$^3$

Now since we only have half a sphere, divide the volume by .

$\text{Volume of $Figure}$=\frac{1}{2}$($\frac{4}{3}$\pi\times$\text{radius}$)^2$

$\text{Volume of $Figure}$=\frac{2}{3}$\pi\times$\text{radius}$^3$

Plug in the given radius to find the volume of the figure.

$$\text{Volume of $Figure}$=\frac{2}{3}$\pi(3)^3$=18\pi=56.55$

Make sure to round to places after the decimal.

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Question

A sphere is cut in half as shown by the figure below.

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

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Answer

Recall how to find the volume of a sphere:

$\text{Volume of $Sphere}$=\frac{4}{3}$\pi\times$\text{radius}$^3$

Now since we only have half a sphere, divide the volume by .

$\text{Volume of $Figure}$=\frac{1}{2}$($\frac{4}{3}$\pi\times$\text{radius}$)^2$

$\text{Volume of $Figure}$=\frac{2}{3}$\pi\times$\text{radius}$^3$

Plug in the given radius to find the volume of the figure.

$$\text{Volume of $Figure}$=\frac{2}{3}$\pi(4)^3$=\frac{128}{3}$\pi=134.04$

Make sure to round to places after the decimal.

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Question

A sphere is cut in half as shown by the figure below.

If the radius of the sphere is $\frac{1}{2}$ , what is the volume of the figure?

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Answer

Recall how to find the volume of a sphere:

$\text{Volume of $Sphere}$=\frac{4}{3}$\pi\times$\text{radius}$^3$

Now since we only have half a sphere, divide the volume by .

$\text{Volume of $Figure}$=\frac{1}{2}$($\frac{4}{3}$\pi\times$\text{radius}$)^2$

$\text{Volume of $Figure}$=\frac{2}{3}$\pi\times$\text{radius}$^3$

Plug in the given radius to find the volume of the figure.

$$\text{Volume of $Figure}$=\frac{2}{3}$\pi($\frac{1}{2}$)^3$=\frac{1}{12}$\pi=0.26$

Make sure to round to places after the decimal.

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Question

A sphere is cut in half as shown by the figure below.

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

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Answer

Recall how to find the volume of a sphere:

$\text{Volume of $Sphere}$=\frac{4}{3}$\pi\times$\text{radius}$^3$

Now since we only have half a sphere, divide the volume by .

$\text{Volume of $Figure}$=\frac{1}{2}$($\frac{4}{3}$\pi\times$\text{radius}$)^2$

$\text{Volume of $Figure}$=\frac{2}{3}$\pi\times$\text{radius}$^3$

Plug in the given radius to find the volume of the figure.

$$\text{Volume of $Figure}$=\frac{2}{3}$\pi(5)^3$=\frac{250}{3}$\pi=261.80$

Make sure to round to places after the decimal.

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Question

A sphere is cut in half as shown by the figure below.

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

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Answer

Recall how to find the volume of a sphere:

$\text{Volume of $Sphere}$=\frac{4}{3}$\pi\times$\text{radius}$^3$

Now since we only have half a sphere, divide the volume by .

$\text{Volume of $Figure}$=\frac{1}{2}$($\frac{4}{3}$\pi\times$\text{radius}$)^2$

$\text{Volume of $Figure}$=\frac{2}{3}$\pi\times$\text{radius}$^3$

Plug in the given radius to find the volume of the figure.

$$\text{Volume of $Figure}$=\frac{2}{3}$\pi(6)^3$=144\pi=452.39$

Make sure to round to places after the decimal.

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Question

A sphere is cut in half as shown by the figure below.

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

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Answer

Recall how to find the volume of a sphere:

$\text{Volume of $Sphere}$=\frac{4}{3}$\pi\times$\text{radius}$^3$

Now since we only have half a sphere, divide the volume by .

$\text{Volume of $Figure}$=\frac{1}{2}$($\frac{4}{3}$\pi\times$\text{radius}$)^2$

$\text{Volume of $Figure}$=\frac{2}{3}$\pi\times$\text{radius}$^3$

Plug in the given radius to find the volume of the figure.

$$\text{Volume of $Figure}$=\frac{2}{3}$\pi(7)^3$=\frac{686}{3}$\pi=718.38$

Make sure to round to places after the decimal.

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Question

A sphere is cut in half as shown by the figure below.

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

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Answer

Recall how to find the volume of a sphere:

$\text{Volume of $Sphere}$=\frac{4}{3}$\pi\times$\text{radius}$^3$

Now since we only have half a sphere, divide the volume by .

$\text{Volume of $Figure}$=\frac{1}{2}$($\frac{4}{3}$\pi\times$\text{radius}$)^2$

$\text{Volume of $Figure}$=\frac{2}{3}$\pi\times$\text{radius}$^3$

Plug in the given radius to find the volume of the figure.

$$\text{Volume of $Figure}$=\frac{2}{3}$\pi(8)^3$=\frac{1024}{3}$\pi=1072.33$

Make sure to round to places after the decimal.

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