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PSAT Math : PSAT Mathematics

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Example Questions

Example Question #1 : Cylinders

A water glass has the shape of a right cylinder. The glass has an interior radius of 2 inches, and a height of 6 inches. The glass is 75% full. What is the volume of the water in the glass (in cubic inches)?

Possible Answers:

$9 \pi$

$12 \pi$

$18 \pi$

$24 \pi$

$36 \pi$

Correct answer:

$18 \pi$

Explanation:

The volume of a right cylinder with radius r=2 and height h=6 is:

$\pi r^2 h = \pi (2)^2 (6) = 24 \pi$

Since the glass is only 75% full, only 75% of the interior volume of the glass is occupied by water. Therefore the volume of the water is:

$24 \pi \times ( 75 / 100 ) = 24 \pi (0.75) = 18 \pi$

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Example Question #23 : Cylinders

A circle has a circumference of $4\pi$ and it is used as the base of a cylinder. The cylinder has a surface area of $16\pi$ . Find the volume of the cylinder.

Possible Answers:

$8\pi$

$10\pi$

$4\pi$

$6\pi$

$2\pi$

Correct answer:

$8\pi$

Explanation:

Using the circumference, we can find the radius of the circle. The equation for the circumference is $2\pi r$ ; therefore, the radius is 2.

Now we can find the area of the circle using $\pi r^{2}$ . The area is $4\pi$ .

Finally, the surface area consists of the area of two circles and the area of the mid-section of the cylinder: $2\cdot 4\pi +4\pi h=16\pi$ , where $h$ is the height of the cylinder.

Thus, $h=2$ and the volume of the cylinder is $4\pi h=4\pi \cdot 2=8\pi$ .

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Example Question #311 : Psat Mathematics

A metal cylindrical brick has a height of $6\ in$ . The area of the top is $25\pi\ in^2$ . A circular hole with a radius of $2\ in$ is centered and drilled half-way down the brick. What is the volume of the resulting shape?

Possible Answers:

$150\pi\ in^3$

$175\pi\ in^3$

$138\pi\ in^3$

$100\pi\ in^3$

$125\pi\ in^3$

Correct answer:

$138\pi\ in^3$

Explanation:

To find the final volume, we will need to subtract the volume of the hole from the total initial volume of the cylinder.

The volume of a cylinder is given by the product of the base area times the height: $V=A_{base}h$ .

Find the initial volume using the given base area and height.

$V_i=(25\pi)(6)=150\pi\ in^3$

Next, find the volume of the hole that was drilled. The base area of this cylinder can be calculated from the radius of the hole. Remember that the height of the hole is only half the height of the block.

$V_{hole}=(\pi r^2)h$

$V_{hole}=(\pi(2)^2)(3)=(4\pi)(3)=12\pi\ in^3$

Finally, subtract the volume of the hole from the total initial volume.

$150\pi-12\pi=138\pi\ in^3$

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Example Question #312 : Psat Mathematics

What is the volume of a cylinder with a diameter of 13 inches and a height of 27.5 inches?

Possible Answers:

$381.44\pi \ in^{3}$

$357.5\pi \ in^{3}$

$2583.1\pi \ in^{3}$

$4647.5\ in^{3}$

$1161.88\pi \ in^{3}$

Correct answer:

$1161.88\pi \ in^{3}$

Explanation:

The equation for the volume of a cylinder is V = Ah, where A is the area of the base and h is the height.

Thus, the volume can also be expressed as V = πr²h.

The diameter is 13 inches, so the radius is 13/2 = 6.5 inches.

Now we can easily calculate the volume:

V = 6.5²π * 27.5 = 1161.88π in³

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Example Question #311 : Geometry

A water tank takes the shape of a closed cylinder whose exterior has height 40 feet and a base with radius 12 feet; the tank is six inches thick throughout. To the nearest hundred, how many cubic feet of water does the tank hold?

Possible Answers:

$14,800 \textrm{ ft}^{3}$

$16,200 \textrm{ ft}^{3}$

$5,200 \textrm{ ft}^{3}$

$3,700 \textrm{ ft}^{3}$

$16,400 \textrm{ ft}^{3}$

Correct answer:

$16,200 \textrm{ ft}^{3}$

Explanation:

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

$40 - 2 \times 0.5 = 39$ feet;

the radius of the base of the interior of the tank is

12 - 0.5 = 11.5 feet.

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

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

$V \approx 3.14 \cdot 11.5^{2} \cdot 39 \approx 16, 195$ cubic feet.

This rounds to 16,200.

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Example Question #22 : How To Find The Volume Of A Cylinder

Barrel

The above diagram is one of a cylindrical tub. The company wants to make a cylindrical tub with four times the volume, but whose base is only three times the radius. How high should this new tub be?

Possible Answers:

$30 \textup{ in}$

$90 \textup{ in}$

$17 \frac{7}{9} \textup{ in}$

$53\frac{1}{3} \textup{ in}$

$22\frac{1}{2} \textup{ in}$

Correct answer:

$17 \frac{7}{9} \textup{ in}$

Explanation:

The volume of the given tub can be expressed using the following formula, setting r = 25 and h= 40 ,

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

$V = \pi \cdot 25^{2} \cdot 40$

$V = 25,000 \pi$ cubic inches.

The new tub should have four times this volume, or

$V = 25,000 \pi \cdot 4 = 100,000 \pi$ cubic inches.

The radius is to be three times that of the above tub, which will be

$r = 25 \cdot 3 = 75$ inches.

The height can therefore be calculated as follows:

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

$\pi \cdot75 ^{2} \cdot h = 100,000 \pi$

$5,625\pi h = 100,000\pi$

$h = \frac{100,000 \pi}{5,625 \pi} = 17 \frac{7}{9}$ inches

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Example Question #24 : How To Find The Volume Of A Cylinder

Barrel

The above diagram is one of a cylindrical tub. The tub is holding water at 70% capacity. To the nearest cubic foot, how much more water can it hold?

Possible Answers:

$38 \textup{ ft}^{3}$

$9 \textup{ ft}^{3}$

$29 \textup{ ft}^{3}$

$23 \textup{ ft}^{3}$

$14 \textup{ ft}^{3}$

Correct answer:

$14 \textup{ ft}^{3}$

Explanation:

The volume of the cylinder can be calculated using the following formula, setting r = 25 and h= 40 ,

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

$V \approx 3.14 \cdot 25^{2} \cdot 40 \approx 78,539.8$ cubic inches.

The tub is 70% full, so it is 30% empty; it can hold

$78,539.8 \times 0.30 \approx 23,561.7$ more cubic inches of water.

The problem asks for the number of cubic feet, so divide by the cube of 12, or 1,728:

$23,561.7 \div 1,728 \approx 13.6$

The correct response is 14 cubic feet.

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Example Question #25 : How To Find The Volume Of A Cylinder

$\textup{A cylindrical container with a diameter of 4 cm and a height of 16 cm}$

$\textup{is filled half full with water. What is the volume of the water in the glass?}$

Possible Answers:

$96\pi \: \textup{ cm}^{3}$

$32\pi \: \textup{ cm}^{3}$

$144\pi \: \textup{ cm}^{3}$

$48\pi \: \textup{ cm}^{3}$

$64\pi \: \textup{ cm}^{3}$

Correct answer:

$32\pi \: \textup{ cm}^{3}$

Explanation:

$\textup{Volume}= \pi r^{2}h\; \; \; \;r=2, h = 8 \textup{ (only half full)}$

$\textup{V}=2^{2}\cdot 8\pi \: \: \: \: \: \textup{V}=32\pi$

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Example Question #841 : Sat Mathematics

Two cylinders are full of milk. The first cylinder is 9” tall and has a base diameter of 3”. The second cylinder is 9” tall and has a base diameter of 4”. If you are going to pour both cylinders of milk into a single cylinder with a base diameter of 6”, how tall must that cylinder be for the milk to fill it to the top?

Possible Answers:

30"

6.25"

12"

Correct answer:

6.25"

Explanation:

Volume of cylinder = π * (base radius)² x height = π * (base diameter / 2 )² x height

Volume Cylinder 1 = π * (3 / 2 )² x 9 = π * (1.5 )² x 9 = π * 20.25

Volume Cylinder 2 = π * (4 / 2 )² x 9 = π * (2 )² x 9 = π * 36

Total Volume = π * 20.25 + π * 36

Volume of Cylinder 3 = π * (6 / 2 )² x H = π * (3 )² x H = π * 9 x H

Set Total Volume equal to the Volume of Cylinder 3 and solve for H

π * 20.25 + π * 36 = π * 9 x H

20.25 + 36 = 9 x H

H = (20.25 + 36) / 9 = 6.25”

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Example Question #1 : How To Find The Length Of An Arc

Figure not drawn to scale.

In the figure above, circle C has a radius of 18, and the measure of angle ACB is equal to 100°. What is the perimeter of the red shaded region?

Possible Answers:

18 + 36π

36 + 36π

36 + 20π

36 + 10π

18 + 10π

Correct answer:

36 + 10π

Explanation:

The perimeter of any region is the total distance around its boundaries. The perimeter of the shaded region consists of the two straight line segments, AC and BC, as well as the arc AB. In order to find the perimeter of the whole region, we must add the lengths of AC, BC, and the arc AB.

The lengths of AC and BC are both going to be equal to the length of the radius, which is 18. Thus, the perimeter of AC and BC together is 36.

Lastly, we must find the length of arc AB and add it to 36 to get the whole perimeter of the region.

Angle ACB is a central angle, and it intercepts arc AB. The length of AB is going to equal a certain portion of the circumference. This portion will be equal to the ratio of the measure of angle ACB to the measure of the total degrees in the circle. There are 360 degrees in any circle. The ratio of the angle ACB to 360 degrees will be 100/360 = 5/18. Thus, the length of the arc AB will be 5/18 of the circumference of the circle, which equals 2πr, according to the formula for circumference.

length of arc AB = (5/18)(2πr) = (5/18)(2π(18)) = 10π.

Thus, the length of arc AB is 10π.

The total length of the perimeter is thus 36 + 10π.

The answer is 36 + 10π.

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PSAT Math : PSAT Mathematics

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #1 : Cylinders

Example Question #23 : Cylinders

Example Question #311 : Psat Mathematics

Example Question #312 : Psat Mathematics

Example Question #311 : Geometry

Example Question #22 : How To Find The Volume Of A Cylinder

Example Question #24 : How To Find The Volume Of A Cylinder

Example Question #25 : How To Find The Volume Of A Cylinder

Example Question #841 : Sat Mathematics

Example Question #1 : How To Find The Length Of An Arc

All PSAT Math Resources

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