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PSAT Math : Geometry

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Example Questions

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

A sphere is perfectly contained within a cube that has a surface area of 726 square units. In square units, what is the surface area of the sphere?

Possible Answers:

484π

None of the other answers

30.25π

11π

121π

Correct answer:

121π

Explanation:

To begin, we must determine the dimensions of the cube. To do this, recall that the surface area of a cube is made up of six squares and is thus defined as: A = 6s², where s is one of the sides of the cube. For our data, this gives us:

726 = 6s²; 121 = s²; s = 11

Now, if the sphere is contained within the cube, that means that 11 represents the diameter of the sphere. Therefore, the radius of the sphere is 5.5 units. The surface area of a sphere is defined as: A = 4πr². For our data, that would be:

A = 4π * 5.5² = 30.25 * 4 * π = 121π

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

The area of a circle with radius 4 divided by the surface area of a sphere with radius 2 is equal to:

Possible Answers:

0.5

Correct answer:

Explanation:

The surface area of a sphere is 4πr². The area of a circle is πr². 16/16 is equal to 1.

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Example Question #13 : Spheres

What is the ratio of the surface area of a cube to the surface area of a sphere inscribed within it?

Possible Answers:

6/π

2π

π/3

3/π

4/π

Correct answer:

6/π

Explanation:

Let's call the radius of the sphere r. The formula for the surface area of a sphere (A) is given below:

A = 4πr²

Because the sphere is inscribed inside the cube, the diameter of the sphere is equal to the side length of the cube. Because the diameter is twice the length of the radius, the diameter of the sphere is 2r. This means that the side length of the cube is also 2r.

The surface area for a cube is given by the following formula, where s represents the length of each side of the cube:

surface area of cube = 6s²

The formula for surface area of a cube comes from the fact that each face of the cube has an area of s², and there are 6 faces total on a cube.

Since we already determined that the side length of the cube is the same as 2r, we can replace s with 2r.

surface area of cube = 6(2r)²= 6(2r)(2r) = 24r².

We are asked to find the ratio of the surface area of the cube to the surface area of the sphere. This means we must divide the surface area of the cube by the surface area of the sphere.

ratio = (24r²)/(4πr²)

The r²term cancels in the numerator and denominator. Also, 24/4 simplifes to 6.

ratio = (24r²)/(4πr²) = 6/π

The answer is 6/π.

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Example Question #191 : Geometry

What is the surface area of a hemisphere with a diameter of $4\ cm$ ?

Possible Answers:

$20\pi \ cm^{2}$

$12\pi \ cm^{2}$

$8\pi \ cm^{2}$

$16\pi \ cm^{2}$

$18\pi \ cm^{2}$

Correct answer:

$12\pi \ cm^{2}$

Explanation:

A hemisphere is half of a sphere. The surface area is broken into two parts: the spherical part and the circular base.

The surface area of a sphere is given by $SA = 4\pi r^{2}$ .

So the surface area of the spherical part of a hemisphere is $SA = 2\pi r^{2}$ .

The area of the circular base is given by $A = \pi r^{2}$ . The radius to use is half the diameter, or 2 cm.

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Example Question #191 : Psat Mathematics

If a sphere has an approximate volume of $904.32\textup{ cubic meters}$ , then what is the approximate diameter of this sphere?

Possible Answers:

$17\textup{m}$

$9\textup{m}$

$12\textup{m}$

$10\textup{m}$

$15\textup{m}$

Correct answer:

$12\textup{m}$

Explanation:

The formula for the volume of a sphere is

$v=\left ( \frac{4}{3} \right )\pi r^3$

Therefore,

$904.32=\left ( \frac{4}{3} \right )\pi r^3$

Dividing both sides by $\left ( \frac{4}{3} \right )\pi$ , leaves us with $r^3\approx215.89$ . Taking the cube root, we find $r\approx5.999$ , meaning our diameter $d\approx12$ .

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

A cube with volume 27 cubic inches is inscribed inside a sphere such that each vertex of the cube touches the sphere. What is the radius, in inches, of the sphere?

Possible Answers:

√3/2 (approximately 1.73)

(3√3)/2 (approximately 2.60)

8.5

Correct answer:

(3√3)/2 (approximately 2.60)

Explanation:

We know that the cube has a volume of 27 cubic inches, so each side of the cube must be ∛27=3 inches. Since the cube is inscribed inside the sphere, the diameter of the sphere is the diagonal length of the cube, so the radius of the sphere is half of the diagonal length of the cube. To find the diagonal length of the cube, we use the distance formula d=√(3²+3²+3² )=√(3*3² )=3√3, and then divide the result by 2 to find the radius of the sphere, (3√3)/2.

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

The surface area of a sphere is 100π square feet. What is the radius in feet?

Possible Answers:

100

Correct answer:

Explanation:

S = 4π(r²)

100π = 4π(r²)

100 = 4r²

25 = r²

5 = r

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

A spherical water tank has a surface area of 400 square meters. To the nearest tenth of a meter, give the radius of the tank.

Possible Answers:

$11.3\textup{ m}$

$5.6\textup{ m}$

$9.1\textup{ m}$

$4.6\textup{ m}$

$3.2\textup{ m}$

Correct answer:

$5.6\textup{ m}$

Explanation:

Given radius , the surface area of a sphere is given by the formula

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

Set S = 400 and solve for :

$4 \pi r^{2} = 400$

$4 \pi r^{2}\div 4 = 400 \div 4$

$\pi r^{2} = 100$

$\pi r^{2} \div \pi = 100 \div \pi$

$r^{2} \approx 100 \div 3.14159 \approx 31.831$

$r \approx \sqrt{31.831} \approx 5.6$ meters.

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Example Question #4 : How To Find The Radius Of A Sphere

A spherical tank is to hold 50,000 liters of water. Given that one cubic meter is equal to 1,000 liters, give the radius of this tank to the nearest tenth of a meter.

Possible Answers:

$2.3\textup{ m}$

$3.5\textup{ m}$

$4.0 \textup{ m}$

$4.6\textup{ m}$

$2.0 \textup{ m}$

Correct answer:

$2.3\textup{ m}$

Explanation:

Given radius , the volume of a sphere is given by the formula

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

Since 1,000 liters = 1 cubic meter, 50,000 liters = 50 cubic meters, and we find by setting V= 50 :

$\frac{4}{3} \pi r^{3} = 50$

$\frac{3}{4} \cdot \frac{4}{3} \pi r^{3} = \frac{3}{4} \cdot 50$

$\pi r^{3} = 37.5$

$r^{3} =\frac{ 37.5}{\pi} \approx 11.9366$

$r \approx\sqrt[3]{ 11.9366} \approx 2.3$ meters.

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Example Question #4 : How To Find The Radius Of A Sphere

The Kelvin temperature scale is basically the same as the Celsius scale except with a different zero point; to convert degrees Celsius to Kelvins, add 273. Also, by Charles's law, the volume of a given mass of gas varies directly as its temperature, expressed in Kelvins.

In the early morning, when the temperature is $5^{\circ }C$ , a spherical balloon is filled with helium until its radius is 100 meters. At 2:00 PM, the temperature is $20^{\circ }C$ . To the nearest tenth of a meter, what is the radius of the balloon at this time?

Possible Answers:

$98.3 \textup{ m}$

$107.2\textup{ m}$

$101.8 \textup{ m}$

$63.0\textup{ m}$

$158.7 \textup{ m}$

Correct answer:

$101.8 \textup{ m}$

Explanation:

The volume of the balloon is given by the formula

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

Also, by the variation relationship,

$\frac{V'}{T'} = \frac{V}{T}$

We do not need to calculate the original volume of the balloon; we can simply replace the volume with the formula:

$\frac{ \frac{4}{3} \pi r'^{3}}{T'} = \frac{ \frac{4}{3} \pi r^{3}}{T}$

Dividing both sides by $\frac{4}{3} \pi$ yields a new equation:

$\frac{ r'^{3}}{T'} = \frac{ r^{3}}{T}$

The initial radius is r = 100

The initial temperature is $T = 273 + 5 = 278 \textup{ K}$

The final (noon) temperature is $T ' = 273 + 20= 293 \textup{ K}$

Substitute to find the final radius:

$\frac{ r'^{3}}{293} = \frac{ 100^{3}}{278}$

$r'^{3} = \frac{ 1,000,000}{278} \cdot 293 \approx 1,053,957$

$r' = \approx\sqrt[3]{ 1,053,957} \approx 101.8$ meters.

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PSAT Math : Geometry

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

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

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

Example Question #13 : Spheres

Example Question #191 : Geometry

Example Question #191 : Psat Mathematics

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

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

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

Example Question #4 : How To Find The Radius Of A Sphere

Example Question #4 : How To Find The Radius Of A Sphere

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