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PSAT Math : How to find the volume of a sphere

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Example Question #181 : Psat Mathematics

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?

Possible Answers:

$\frac{4r}{3}$

$\frac{9r}{2}$

$\frac{9}{2r}$

$\frac{3}{r}$

$3r$

Correct answer:

$\frac{9}{2r}$

Explanation:

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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Example Question #181 : Psat Mathematics

A water tank takes the shape of a sphere whose exterior has radius 16 feet; the tank is six inches thick throughout. To the nearest hundred, how many cubic feet of water does the tank hold?

Possible Answers:

$14,100\textrm{ ft}^{3}$

$2,800\textrm{ ft}^{3}$

$15,600\textrm{ ft}^{3}$

$17,100\textrm{ ft}^{3}$

$3,100\textrm{ ft}^{3}$

Correct answer:

$15,600\textrm{ ft}^{3}$

Explanation:

Six inches is equal to 0.5 feet, so the radius of the interior of the tank is

16-0.5 = 15.5 feet.

The amount of water the tank holds is the volume of the interior of the tank, which is

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

$V \approx \frac{4}{3} \cdot 3.14 \cdot 15.5^{3} \approx 15,591$ ,

which rounds to 15,600 cubic feet.

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PSAT Math : How to find the volume of a sphere

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Example Question #181 : Psat Mathematics

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