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SSAT Upper Level Math : How to find the volume of a sphere

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Example Questions

Example Question #1 : Volume Of A Three Dimensional Figure

A car dealership wants to fill a large spherical advertising ballon with helium. It can afford to buy 1,000 cubic yards of helium to fill this balloon. What is the greatest possible diameter of that balloon (nearest tenth of a yard)?

Possible Answers:

$6.2 \;\textrm{yd}$ $\displaystyle 6.2 \;\textrm{yd}$

$9.3 \;\textrm{yd}$ $\displaystyle 9.3 \;\textrm{yd}$

$12.4 \;\textrm{yd}$ $\displaystyle 12.4 \;\textrm{yd}$

$7.7 \;\textrm{yd}$ $\displaystyle 7.7 \;\textrm{yd}$

$15.4 \;\textrm{yd}$ $\displaystyle 15.4 \;\textrm{yd}$

Correct answer:

$12.4 \;\textrm{yd}$ $\displaystyle 12.4 \;\textrm{yd}$

Explanation:

The volume of a sphere, given its radius, is

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

Set $\displaystyle V = 1,000$ , solve for $\displaystyle r$ , and double that to get the diameter.

$1,000= \frac{4\pi r^{3}}{3}$ $\displaystyle 1,000= \frac{4\pi r^{3}}{3}$

$r^{3} = \frac{1,000 \cdot 3 }{4\pi} \approx 238.7$ $\displaystyle r^{3} = \frac{1,000 \cdot 3 }{4\pi} \approx 238.7$

$r \approx \sqrt[3]{238.7} \approx 6.2$ $\displaystyle r \approx \sqrt[3]{238.7} \approx 6.2$

The diameter is twice this, or 12.4 yards.

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Example Question #2 : How To Find The Volume Of A Sphere

The diameter of a sphere is $\displaystyle 6t$ . Give the volume of the sphere in terms of $\displaystyle t$ .

Possible Answers:

$36\pi t^3$ $\displaystyle 36\pi t^3$

$30\pi t^3$ $\displaystyle 30\pi t^3$

$10\pi t^3$ $\displaystyle 10\pi t^3$

$24\pi t^3$ $\displaystyle 24\pi t^3$

$72\pi t^3$ $\displaystyle 72\pi t^3$

Correct answer:

$36\pi t^3$ $\displaystyle 36\pi t^3$

Explanation:

The diameter of a sphere is $\displaystyle 6t$ so the radius of the sphere would be $6t\div 2=3t$ $\displaystyle 6t\div 2=3t$

The volume enclosed by a sphere is given by the formula:

$Volume=\frac{4}{3}\pi r^3=\frac{4}{3}\pi \times (3t)^3=\frac{4}{3}\pi\times 27t^3\Rightarrow Volume=36\pi t^3$ $\displaystyle Volume=\frac{4}{3}\pi r^3=\frac{4}{3}\pi \times (3t)^3=\frac{4}{3}\pi\times 27t^3\Rightarrow Volume=36\pi t^3$

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Example Question #3 : How To Find The Volume Of A Sphere

A spherical balloon has a diameter of 10 meters. Give the volume of the balloon.

Possible Answers:

$\displaystyle 1023.3m^3$

$\displaystyle 723.3m^3$

$\displaystyle 523.6m^3$

500m^3 $\displaystyle 500m^3$

1000m^3 $\displaystyle 1000m^3$

Correct answer:

$\displaystyle 523.6m^3$

Explanation:

The volume enclosed by a sphere is given by the formula:

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

where $\displaystyle r$ is the radius of the sphere. The diameter of the balloon is 10 meters so the radius of the sphere would be $10\div 2=5$ $\displaystyle 10\div 2=5$ meters. Now we can get:

$Volume=\frac{4}{3}\pi r^3=\frac{4}{3}\pi\times 5^3=\frac{4}{3}\times \pi \times 125\approx 523.6 m^3$ $\displaystyle Volume=\frac{4}{3}\pi r^3=\frac{4}{3}\pi\times 5^3=\frac{4}{3}\times \pi \times 125\approx 523.6 m^3$

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Example Question #1 : How To Find The Volume Of A Sphere

The volume of a sphere is 1000 cubic inches. What is the diameter of the sphere.

Possible Answers:

$10\ inches$ $\displaystyle 10\ inches$

$16.4\ inches$ $\displaystyle 16.4\ inches$

$12.4\ inches$ $\displaystyle 12.4\ inches$

$10.4\ inches$ $\displaystyle 10.4\ inches$

$12\ inches$ $\displaystyle 12\ inches$

Correct answer:

$12.4\ inches$ $\displaystyle 12.4\ inches$

Explanation:

The volume of a sphere is:

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

Where $\displaystyle r$ is the radius of the sphere. We know the volume and can solve the formula for $\displaystyle r$ :

$Volume=\frac{4}{3}\pi r^3=1000\Rightarrow r^3=\frac{3\times1000 }{4\pi}\approx 238.85\Rightarrow r\approx 6.2$ $\displaystyle Volume=\frac{4}{3}\pi r^3=1000\Rightarrow r^3=\frac{3\times1000 }{4\pi}\approx 238.85\Rightarrow r\approx 6.2$ inches

So we can get:

$Diameter=2r=2\times 6.2=12.4inches$ $\displaystyle Diameter=2r=2\times 6.2=12.4inches$

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Example Question #2 : Volume Of A Three Dimensional Figure

A sphere has a diameter of $\displaystyle 10$ inches. What is the volume of this sphere?

Possible Answers:

104.7 $\displaystyle 104.7$ in^3 $\displaystyle in^3$

523.6 $\displaystyle 523.6$ in^3 $\displaystyle in^3$

458.3 $\displaystyle 458.3$ in^3 $\displaystyle in^3$

632.9 $\displaystyle 632.9$ in^3 $\displaystyle in^3$

Correct answer:

523.6 $\displaystyle 523.6$ in^3 $\displaystyle in^3$

Explanation:

To find the volume of a sphere, use the following formula:

$\text{Volume}=\frac{4}{3}\pi\times r^3$ $\displaystyle \text{Volume}=\frac{4}{3}\pi\times r^3$ , where $\displaystyle r$ is the radius of the sphere.

Now, because we are given the diameter of the sphere, divide that value in half to find the radius.

$r=10\div2=5$ $\displaystyle r=10\div2=5$ $\displaystyle in$

Now, plug this value into the volume equation.

$\text{Volume}=\frac{4}{3}\pi\times 5^3$ $\displaystyle \text{Volume}=\frac{4}{3}\pi\times 5^3$

$\text{Volume}=\frac{4}{3}\pi\times 125$ $\displaystyle \text{Volume}=\frac{4}{3}\pi\times 125$

$\text{Volume}=\frac{500}{3}\pi=523.6$ $\displaystyle \text{Volume}=\frac{500}{3}\pi=523.6$ in^3 $\displaystyle in^3$

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Example Question #2 : Volume Of A Three Dimensional Figure

What is the volume of a sphere with a diameter of $\displaystyle 10$ ?

Possible Answers:

$\frac{100}{3}\pi$ $\displaystyle \frac{100}{3}\pi$

$200\pi$ $\displaystyle 200\pi$

$100\pi$ $\displaystyle 100\pi$

$150\pi$ $\displaystyle 150\pi$

$\frac{500}{3}\pi$ $\displaystyle \frac{500}{3}\pi$

Correct answer:

$\frac{500}{3}\pi$ $\displaystyle \frac{500}{3}\pi$

Explanation:

Write the formula for the volume of the sphere.

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

The radius is half the diameter, which is five. Substitute the value.

$V=\frac{4}{3}\pi r^3=\frac{4}{3}\pi (5)^3=\frac{4}{3}\pi (125)= \frac{500}{3}\pi$ $\displaystyle V=\frac{4}{3}\pi r^3=\frac{4}{3}\pi (5)^3=\frac{4}{3}\pi (125)= \frac{500}{3}\pi$

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Example Question #1 : Volume Of A Three Dimensional Figure

What is the volume of a sphere with diameter 12 feet?

Possible Answers:

$904.8 ft^{3}$ $\displaystyle 904.8 ft^{3}$

$452.4\;ft^{3}$ $\displaystyle 452.4\;ft^{3}$

$3,619.1\;ft^{2}$ $\displaystyle 3,619.1\;ft^{2}$

None of the other answers are correct

$7,238.2\;ft^{3}$ $\displaystyle 7,238.2\;ft^{3}$

Correct answer:

$904.8 ft^{3}$ $\displaystyle 904.8 ft^{3}$

Explanation:

The radius of the sphere is half the diameter, or 6 feet; use the formula

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

Setting r=6 $\displaystyle r=6$ :

$V= \frac{4}{3} \pi \cdot 6^{3} \approx904.8 ft^{3}$ $\displaystyle V= \frac{4}{3} \pi \cdot 6^{3} \approx904.8 ft^{3}$

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SSAT Upper Level Math : How to find the volume of a sphere

Study concepts, example questions & explanations for SSAT Upper Level Math

All SSAT Upper Level Math Resources

Example Questions

Example Question #1 : Volume Of A Three Dimensional Figure

Example Question #2 : How To Find The Volume Of A Sphere

Example Question #3 : How To Find The Volume Of A Sphere

Example Question #1 : How To Find The Volume Of A Sphere

Example Question #2 : Volume Of A Three Dimensional Figure

Example Question #2 : Volume Of A Three Dimensional Figure

Example Question #1 : Volume Of A Three Dimensional Figure

All SSAT Upper Level Math Resources

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