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PSAT Math : Geometry

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Example Questions

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

An 12-inch cube of wood has a cylinder drilled out of it. The cylinder has a radius of 3.75 inches. If the density of the wood is 4 g/in³, what is the mass of the remaining wood after the cylinder is drilled out?

Possible Answers:

3813.3 g

4791.43 g

3193.33 g

2594.11 g

4921.4 g

Correct answer:

4791.43 g

Explanation:

We must calculate our two volumes and subtract them. Following this, we will multiply by the density.

The volume of the cube is very simple: 12 * 12 * 12, or 1728 in³.

The volume of the cylinder is calculated by multiplying the area of its base by its height. The height of the cylinder is 8 inches (the height of the cube through which it is being drilled). Therefore, its volume is πr²h = π * 3.75²* 12 = 168.75π in³.

The volume remaining in the cube after the drilling is: 1728 – 168.75π, or approximately 1728 – 530.1433125 = 1197.8566875 in³. Now, multiply this by 4 to get the mass: (approx.) 4791.43 g.

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

A hollow prism has a base 5 in x 6 in and a height of 10 in. A closed, cylindrical can is placed in the prism. The remainder of the prism is then filled with gel around the cylinder. The thickness of the can is negligible. Its diameter is 4 in and its height is half that of the prism. What is the approximate volume of gel needed to fill the prism?

Possible Answers:

187.73 in³

203.44 in³

237.17 in³

249.73 in³

103.33 in³

Correct answer:

237.17 in³

Explanation:

The general form of our problem is:

Gel volume = Prism volume – Can volume

The prism volume is simple: 5 * 6 * 10 = 300 in³

The volume of the can is found by multiplying the area of the circular base by the height of the can. The height is half the prism height, or 10/2 = 5 in. The area of the base is equal to πr². Note that the prompt has given the diameter. Therefore, the radius is 2, not 4. The base's area is: 2²π = 4π. The total volume is therefore: 4π * 5 = 20π in³.

The gel volume is therefore: 300 – 20π or (approx.) 237.17 in³.

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

A hollow prism has a base 12 in x 13 in and a height of 42 in. A closed, cylindrical can is placed in the prism. The remainder of the prism is then filled with gel, surrounding the can. The thickness of the can is negligible. Its diameter is 9 in and its height is one-fourth that of the prism. The can has a mass of 1.5 g per in³, and the gel has a mass of 2.2 g per in³. What is the approximate overall mass of the contents of the prism?

Possible Answers:

139.44 g

973.44 g

11.48 kg

13.95 kg

15.22 kg

Correct answer:

13.95 kg

Explanation:

We must find both the can volume and the gel volume. The formula for the gel volume is:

Gel volume = Prism volume – Can volume

The prism volume is simple: 12 * 13 * 42 = 6552 in³

The volume of the can is found by multiplying the area of the circular base by the height of the can. The height is one-fourth the prism height, or 42/4 = 10.5 in. The area of the base is equal to πr². Note that the prompt has given the diameter. Therefore, the radius is 4.5, not 9. The base's area is: 4.5²π = 20.25π. The total volume is therefore: 20.25π * 10.5 = 212.625π in³.

The gel volume is therefore: 6552 – 212.625π or (approx.) 5884.02 in³.

The approximate volume for the can is: 667.98 in³

From this, we can calculate the approximate mass of the contents:

Gel Mass = Gel Volume * 2.2 = 5884.02 * 2.2 = 12944.844 g

Can Mass = Can Volume * 1.5 = 667.98 * 1.5 = 1001.97 g

The total mass is therefore 12944.844 + 1001.97 = 13946.814 g, or approximately 13.95 kg.

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Example Question #301 : Geometry

Jessica wishes to fill up a cylinder with water at a rate of gallons per minute. The volume of the cylinder is gallons. The hole at the bottom of the cylinder leaks out 0.5 gallons per minute. If there are gallons in the cylinder when Jessica starts filling it, how long does it take to fill?

Possible Answers:

$11\: \text{min}$

$3\: \text{min}$

$8\: \text{min}$

$5\: \text{min}$

$6\: \text{min}$

Correct answer:

$8\: \text{min}$

Explanation:

Jessica needs to fill up gallons at the effective rate of $1.5\:\text{gal/min}$ . $12\: \text{gal}$ divided by $1.5\: \text{gal/min}$ is equal to $8 \:\text{min}$ . Notice how the units work out.

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Example Question #1362 : Concepts

A vase needs to be filled with water. If the vase is a cylinder that is $\dpi{100} \small 12{}''$ tall with a $\dpi{100} \small 2{}''$ radius, how much water is needed to fill the vase?

Possible Answers:

$\dpi{100} \small 12\pi$

$\dpi{100} \small 48\pi$

$\dpi{100} \small 64\pi$

$\dpi{100} \small 96\pi$

$\dpi{100} \small 24\pi$

Correct answer:

$\dpi{100} \small 48\pi$

Explanation:

Cylinder

$\dpi{100} \small V = \pi r^{2}h$

$\dpi{100} \small V = \pi (2)^{2}\times 12$

$\dpi{100} \small V = 4\times 12\pi$

$\dpi{100} \small V = 48\pi$

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Example Question #21 : Cylinders

A cylinder has a base diameter of 12 in and is 2 in tall. What is the volume?

Possible Answers:

$48\pi in^{3}$

$72\pi in^{3}$

$288\pi in^{3}$

$36\pi in^{3}$

$24\pi in^{3}$

Correct answer:

$72\pi in^{3}$

Explanation:

The volume of a cylinder is $V=\pi r^{2}h$

The diameter is given, so make sure to divide it in half.

The units are inches cubed in this example

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Example Question #11 : How To Find The Volume Of A Cylinder

What is the volume of a cylinder with a radius of 4 and a height of 5?

Possible Answers:

$40\pi$

$60\pi$

$54\pi$

$72\pi$

$80\pi$

Correct answer:

$80\pi$

Explanation:

$volume = \pi r^{2}h = \pi \cdot 4^{2} \cdot 5 = 80\pi$

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Example Question #12 : How To Find The Volume Of A Cylinder

Claire's cylindrical water bottle is 9 inches tall and has a diameter of 6 inches. How many cubic inches of water will her bottle hold?

Possible Answers:

$81\pi$

$45\pi$

$54\pi$

$18\pi$

$324\pi$

Correct answer:

$81\pi$

Explanation:

The volume is the area of the base times the height. The area of the base is $\pi r^{2}$ , and the radius here is 3. $\pi\cdot 3^{2}\cdot 9=81\pi$

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Example Question #13 : Cylinders

What is the volume of a circular cylinder whose height is 8 cm and has a diameter of 4 cm?

Possible Answers:

$12\pi \ cm^{3}$

$64\pi \ cm^{3}$

$128\pi \ cm^{3}$

$32\pi \ cm^{3}$

$16\pi \ cm^{3}$

Correct answer:

$32\pi \ cm^{3}$

Explanation:

The volume of a circular cylinder is given by $V = \pi r^{2}h$ where is the radius and is the height. The diameter is given as 4 cm, so the radius would be 2 cm as the diameter is twice the radius.

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Example Question #832 : Geometry

You have tall glass with a radius of 3 inches and height of 6 inches. You have an ice cube tray that makes perfect cubic ice cubes that have 0.5 inch sides. You put three ice cubes in your glass. How much volume do you have left for soda? The conversion factor is $1\hspace{1 mm}inch^3=0.0163871\hspace{1 mm}L$ .

Possible Answers:

$54\pi\hspace{1 mm}inches^3$

$\frac{3}{8}inches^3$

$2.77\hspace{1 mm}L$

$1.69\hspace{1 mm}L$

$169.27\hspace{1 mm}mL$

Correct answer:

$2.77\hspace{1 mm}L$

Explanation:

First we will calculate the volume of the glass. The volume of a cylinder is

$V=\pi r^2 h$

$V=\pi (3\hspace{1 mm}inches)^2 \times 6\hspace{1 mm}inches = 54\pi\hspace{1 mm}inches^3$

Now we will calculate the volume of one ice cube:

$V=lwh=l^3=(\frac{1}{2}\hspace{1 mm}inch)^3=\frac{1}{8}\hspace{1 mm}inches^3$

The volume of three ice cubes is $\frac{3}{8}\hspace{1 mm}inches^3$ . We will then subtract the volume taken up by ice from the total volume:

$54\pi\hspace{1 mm}inches^3-\frac{1}{8}\hspace{1 mm}inches^3\approx 169.27\hspace{1 mm}inches^3$

Now we will use our conversion factor:

$169.27\hspace{1 mm}inches^3\times \frac{0.016387\hspace{1 mm}L}{1\hspace{1 mm}inch^3}=2.77\hspace{1 mm}L$

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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 Volume Of A Cylinder

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

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

Example Question #301 : Geometry

Example Question #1362 : Concepts

Example Question #21 : Cylinders

Example Question #11 : How To Find The Volume Of A Cylinder

Example Question #12 : How To Find The Volume Of A Cylinder

Example Question #13 : Cylinders

Example Question #832 : Geometry

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