PSAT Math : PSAT Mathematics

Study concepts, example questions & explanations for PSAT Math

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Example Questions

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

A company wants to build a cubical room around a cone so that the cone's height and diameter are 3 inch less than the dimensions of the cube. If the volume of the cone is 486π ft3, what is the surface area of the cube?

Possible Answers:

513.375 in2

486 in2

726 in2

73,926 in2

69,984 in2

Correct answer:

73,926 in2

Explanation:

To begin, we need to solve for the dimensions of the cone.

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

Using our data, we know that h = 2r because the height of the cone matches its diameter (based on the prompt).

486π  = (1/3)πr* 2r = (2/3)πr3

Multiply both sides by (3/2π): 729 = r3

Take the cube root of both sides: r = 9

Note that this is in feet. The answers are in square inches. Therefore, convert your units to inches: 9 * 12 = 108, then add 3 inches to this: 108 + 3 = 111 inches.

The surface area of the cube is defined by: A = 6 * s2, or for our data, A = 6 * 1112 = 73,926 in2

Example Question #11 : Solid Geometry

If the volume of a cube is 216 cubic units, then what is its surface area in square units?

Possible Answers:

36

54

64

216

108

Correct answer:

216

Explanation:

The volume of a cube is given by the formula V = s^{3}, where V is the volume, and s is the length of each side. We can set V to 216 and then solve for s.

\inline 216 = s^{3}

In order to find s, we can find the cube root of both sides of the equaton. Finding the cube root of a number is the same as raising that number to the one-third power.

\sqrt[3]{216}= 216^{1/3}=6=s

This means the length of the side of the cube is 6. We can use this information to find the surface area of the cube, which is equal to \inline 6s^{2}. The formula for surface area comes from the fact that each face of the cube has an area of s^2, and there are 6 faces in a cube.

surface area = 6s^{2}=6(6^{2})=6(36)=216

The surface area of the square is 216 square units.

The answer is 216.

Example Question #12 : Solid Geometry

You have a cube with sides of 4.5 inches. What is the surface area of the cube?

Possible Answers:

Correct answer:

Explanation:

The area of one side of the cube is:

A cube has 6 sides, so the total surface area of the cube is

Example Question #4 : How To Find Surface Area

A cube has a surface area of 24. If we double the height of the cube, what is the volume of the new rectangular box?

Possible Answers:

Correct answer:

Explanation:

We have a cube with a surface area of 24, which means each side has an area of 4. Therefore, the length of each side is 2. If we double the height, the volume becomes .

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

A cube has a surface area of 10m2. If a cube's sides all double in length, what is the new surface area?

Possible Answers:

640m2

40m2

80m2

320m2

20m2

Correct answer:

40m2

Explanation:

The equation for surface area of the original cube is 6s2. If the sides all double in length the new equation is 6(2s)2 or 6 * 4s2. This makes the new surface area 4x that of the old. 4x10 = 40m2

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

If a cube has an area of , then what is the surface area of this cube?

Possible Answers:

Correct answer:

Explanation:

The area of a cube is equal to the measure of one edge cubed. If we take the cube root of , we get , so the edges of this cube measure ; therefore, one face of the cube has an area of  square inches, because .

There are  sides to a cube, so

Example Question #691 : Geometry

If a cube is 3” on all sides, what is the length of the diagonal of the cube?

Possible Answers:

27

9

4√3

3√3

3√2

Correct answer:

3√3

Explanation:

General formula for the diagonal of a cube if each side of the cube = s

Use Pythagorean Theorem to get the diagonal across the base:

s2 + s2 = h2

And again use Pythagorean Theorem to get cube’s diagonal, then solve for d:

h2 + s2 = d2

s2 + s2 + s2 = d2

3 * s2 = d2

d = √ (3 * s2) = s √3

So, if s = 3 then the answer is 3√3

Example Question #1 : Cubes

A cube is inscribed in a sphere of radius 1 such that all 8 vertices of the cube are on the surface of the sphere.  What is the length of the diagonal of the cube?

Possible Answers:

8

2

1

(3)

(2)

Correct answer:

2

Explanation:

Since the diagonal of the cube is a line segment that goes through the center of the cube (and also the circumscribed sphere), it is clear that the diagonal of the cube is also the diameter of the sphere.  Since the radius = 1, the diameter = 2.

Example Question #1 : Solid Geometry

What is the length of the diagonal of a cube with volume of 512 in3?

Possible Answers:

None of the other answers

8√(3) in

4√(3) in

8 in

2√(6) in

Correct answer:

8√(3) in

Explanation:

The first thing necessary is to determine the dimensions of the cube.  This can be done using the volume formula for cubes: V = s3, where s is the length of the cube. For our data, this is:

s3 = 512, or (taking the cube root of both sides), s = 8.

The distance from corner to corner of the cube will be equal to the distance between (0,0,0) and (8,8,8).  The distance formula for three dimensions is very similar to that of 2 dimensions (and hence like the Pythagorean Theorem):

d√( (x1 – x2)2 + (y1 – y2)2  + (z1 – z2)2

Or for our simpler case:

d = √( (x)2 + (y)2  + (z)2) = √( (s)2 + (s)2  + (s)2) = √( (8)2 + (8)2  + (8)2) = √( 64 + 64 + 64) = √(64 * 3) = 8√(3)

Example Question #2 : Solid Geometry

What is the length of the diagonal of a cube with volume of 1728 in3?

Possible Answers:

12√(3) in

3√(3) in

12 in

18 in

6√(3) in

Correct answer:

12√(3) in

Explanation:

The first thing necessary is to determine the dimensions of the cube.  This can be done using the volume formula for cubes: V = s3, where s is the length of the cube. For our data, this is:

s3 = 1728, or (taking the cube root of both sides), s = 12.

The distance from corner to corner of the cube will be equal to the distance between (0,0,0) and (12,12,12).  The distance formula for three dimensions is very similar to that of 2 dimensions (and hence like the Pythagorean theorem):

d = √( (x1 – x2)2 + (y1 – y2)2  + (z1 – z2)2

Or, for our simpler case:

d = √( (x)2 + (y)2  + (z)2) = √( (s)2 + (s)2  + (s)2) = √( (12)2 + (12)2  + (12)2) = √( 144 + 144 + 144) = √(3 * 144) = 12√(3) = 12√(3)

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