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PSAT Math : Geometry

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Example Questions

Example Question #6 : How To Find The Radius Of A Sphere

The city of Washington wants to build a spherical water tank for the town hall. The tank is to have capacity 120 cubic meters of water.

To the nearest tenth, what will the radius of the tank be?

Possible Answers:

$6.1\textup{ m}$

$5.4\textup{ m}$

The correct answer is not given among the other responses.

$10.7\textup{ m}$

$3.1\textup{ m}$

Correct answer:

$3.1\textup{ m}$

Explanation:

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

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

We find the inner radius by setting V= 120 :

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

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

$\pi r^{3} = 90$

$r^{3} =\frac{ 90}{\pi} \approx 28.65$

$r \approx\sqrt[3]{ 28.65} \approx 3.1$ meters.

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Example Question #2 : 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.

A spherical balloon is filled with 10,000 cubic meters of gas in the morning when the temperature is $10^{\circ }C$ . The temperature increases to $25^{\circ }C$ by noon, with no other conditions changing. What is the radius of the balloon at noon, to the nearest tenth of a meter?

Possible Answers:

$9.8\textup { m}$

$13.1\textup{ m}$

None of the other responses gives the correct answer.

$18.1\textup{ m}$

$13.6 \textup{ m}$

Correct answer:

$13.6 \textup{ m}$

Explanation:

First, we use the variation equation to figure out the volume of the balloon at noon. First, we add 273 to each of the temperatures.

The initial temperature is $273 + 10 = 283 \textup{ K}$

The final (noon) temperature is $273 + 25= 298 \textup{ K}$

Since volume varies directly as temperature, we can set up the equation

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

$\frac{V'}{298} = \frac{10,000}{283}$

$V' = \frac{10,000}{283} \cdot 298 \approx 10,530$

The volume of a sphere is given by the formula

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

so set V = 10,530 and solve for :

$\frac{4}{3} \pi r^{3} = 10,530$

$\frac{3}{4} \cdot \frac{4}{3} \pi r^{3} = \frac{3}{4} \cdot 10,530$

$\pi r^{3} \approx 7,897.5$

$r^{3} \approx 7,897.5 \div \pi \approx 2,513.8$

$r \approx\sqrt[3]{ 2,513.8} \approx 13.6$ meters.

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

Find the radius of a sphere whose volume is $\small \small 36\pi$ $\text{in}^3$ .

Possible Answers:

$\small 9$ $\text{in}$

$\small 9\pi$ $\text{in}$

$\small 3\pi$ $\text{in}$

$\small 3$ $\text{in}$

Correct answer:

$\small 3$ $\text{in}$

Explanation:

Use the equation for the volume of a sphere to find the radius.

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

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

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

$\small 27 = r^{3}$

$\small r = 3$

So, the radius of the sphere is 3

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

If a sphere has a volume of $36\pi\textup{ cubic inches}$ , then what is the radius of the sphere?

Possible Answers:

$6\textup{ in}$

$4.5\textup{ in}$

$3\textup{ in}$

$9\textup{ in}$

$18\textup{ in}$

Correct answer:

$3\textup{ in}$

Explanation:

The volume of a sphere is equal to

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

Therefore,

$r=\sqrt[3]{(3/4)v/\pi}=\sqrt[3]{(3/4)36\pi/\pi}=\sqrt[3]{27}=3$

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Example Question #1 : Cones

An empty tank in the shape of a right solid circular cone has a radius of r feet and a height of h feet. The tank is filled with water at a rate of w cubic feet per second. Which of the following expressions, in terms of r, h, and w, represents the number of minutes until the tank is completely filled?

Possible Answers:

π(r²)(h)/(180w)

180w/(π(r²)(h))

π(r²)(h)/(60w)

π(r²)(h)/(20w)

20w/(π(r²)(h))

Correct answer:

π(r²)(h)/(180w)

Explanation:

The volume of a cone is given by the formula V = (πr²)/3. In order to determine how many seconds it will take for the tank to fill, we must divide the volume by the rate of flow of the water.

time in seconds = (πr²)/(3w)

In order to convert from seconds to minutes, we must divide the number of seconds by sixty. Dividing by sixty is the same is multiplying by 1/60.

(πr²)/(3w) * (1/60) = π(r²)(h)/(180w)

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

A cone has a base radius of 13 in and a height of 6 in. What is its volume?

Possible Answers:

None of the other answers

1352π in³

338π in³

1014π in³

4394π in³

Correct answer:

338π in³

Explanation:

The basic form for the volume of a cone is:

V = (1/3)πr²h

For this simple problem, we merely need to plug in our values:

V = (1/3)π13²* 6 = 169 * 2π = 338π in³

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Example Question #3 : Cones

A cone has a base circumference of 77π in and a height of 2 ft. What is its approximate volume?

Possible Answers:

8893.5π in³

142,296π in³

11,858π in³

71,148π in³

2964.5π in³

Correct answer:

11,858π in³

Explanation:

There are two things to be careful with here. First, we must solve for the radius of the base. Secondly, note that the height is given in feet, not inches. Notice that all the answers are in cubic inches. Therefore, it will be easiest to convert all of our units to inches.

First, solve for the radius, recalling that C = 2πr, or, for our values 77π = 2πr. Solving for r, we get r = 77/2 or r = 38.5.

The height, in inches, is 24.

The basic form for the volume of a cone is: V = (1 / 3)πr²h

For our values this would be:

V = (1/3)π * 38.5² * 24 = 8 * 1482.25π = 11,858π in³

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

What is the volume of a right cone with a diameter of 6 cm and a height of 5 cm?

Possible Answers:

$120\pi \ cm^{3}$

$180\pi \ cm^{3}$

$60\pi \ cm^{3}$

$45\pi \ cm^{3}$

$15\pi \ cm^{3}$

Correct answer:

$15\pi \ cm^{3}$

Explanation:

The general formula is given by $V = 1/3Bh = 1/3\pi r^{2}h$ , where = radius and = height.

The diameter is 6 cm, so the radius is 3 cm.

$V=\frac{\pi 3^2\times 5}{3}=15\pi$

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Example Question #201 : Geometry

There is a large cone with a radius of 4 meters and height of 18 meters. You can fill the cone with water at a rate of 3 cubic meters every 25 seconds. How long will it take you to fill the cone?

Possible Answers:

$600\ \textup{minutes}$

$3\ \textup{minutes}\ 20\ \textup{seconds}$

$41\ \textup{minutes}\ 53\ \textup{seconds}$

$800\ \textup{minutes}$

$60\ \textup{minutes}$

Correct answer:

$41\ \textup{minutes}\ 53\ \textup{seconds}$

Explanation:

First we will calculate the volume of the cone

$V=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi\cdot 4^{2}\cdot 18=96\pi\ m^{3}$

Next we will determine the time it will take to fill that volume

$96\pi \ m^{3}\times \frac{25\ s}{3\ m^{3}}=800\pi \ s$

We will then convert that into minutes

$800\pi \ s\times \frac{1\ minute}{60\ seconds}=13.\overline{3}\pi \ minutes\approx 41.9\ minutes\approx 41\ minutes\ 53\ seconds$

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Example Question #2 : Cones

Cone

Which of these answers comes closest to the volume of the above cone?

Possible Answers:

$0.48\textup{ m}^{3}$

$0.34\textup{ m}^{3}$

$0.29 \textup{ m}^{3}$

$0.44\textup{ m}^{3}$

$0.32\textup{ m}^{3}$

Correct answer:

$0.29 \textup{ m}^{3}$

Explanation:

The radius and the height of a cone are required in order to find its volume.

The radius is 50 centimeters, which can be converted to meters by dividing by 100:

$50 \div 100 = 0.5$ meters

The slant height is 120 centimeters, which converts similarly to

$120 \div 100 = 1.2$ meters

To find the height, we need to use the Pythagorean Theorem with the radius 0.5 as one leg and the slant height 1.2 as the hypotenuse of a right triangle, and the height as the other leg:

$h = \sqrt{1.2^{2} -0.5^{2}} = \sqrt{1.44 -0.25} = \sqrt{1.19} \approx 1.09$ meters.

The volume formula can now be used:

$V = \frac{\pi r^{2}h}{3}$

$\approx \frac{3.14 \cdot 0.5^{2} \cdot 1.09}{3}$

$\approx \frac{3.14 \cdot 0.25 \cdot 1.09}{3}$

$\approx \frac{0.8557}{3}$

$\approx 0.29$ cubic meters

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PSAT Math : Geometry

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #6 : How To Find The Radius Of A Sphere

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

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

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

Example Question #1 : Cones

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

Example Question #3 : Cones

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

Example Question #201 : Geometry

Example Question #2 : Cones

All PSAT Math Resources

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