All PSAT Math Resources
Example Questions
Example Question #31 : Cubes
Kim from Idaho can only stack bales of hay in her barn for 3 hours before she needs a break. She stacks the bales at a rate of 2 bales per minute, 3 bales high with 5 bales in a single row. How many full rows will she have at the end of her stacking?
15
27
24
20
16
24
She will stack 360 bales in 3 hours. One row requires 15 bales. 360 divided by 15 is 24.
Example Question #1 : Cubes
A cube has a volume of . What is the volume of cube with sides that are twice as long?
The volume of a cube is .
If each side of the cube is , then the volume will be .
If we double each side, then each side would be and the volume would be .
Example Question #31 : Solid Geometry
How many smaller boxes with a dimensions of 1 by 5 by 5 can fit into cube shaped box with a surface area of 150?
8
4
7
6
5
5
There surface are of a cube is 6 times the area of one face of the cube , therefore
a is equal to an edge of the cube
the volume of the cube is
The problem states that the dimensions of the smaller boxes are 1 x 5 x 5, the volume of one of the smaller boxes is 25.
Therefore, 125/25 = 5 small boxes
Example Question #1 : How To Find The Volume Of A Cube
If a cube has its edges increased by a factor of 5, what is the ratio of the new volume to the old volume?
A cubic volume is . Let the original sides be 1, so that the original volume is 1. Then find the volume if the sides measure 5. This new volume is 125. Therefore, the ratio of new volume to old volume is 125: 1.
Example Question #32 : Solid Geometry
A cube is inscribed inside a sphere of radius 1 such that each of the eight vertices of the cube lie on the surface of the sphere. What is the volume of the cube?
To make this problem easier to solve, we can inscribe a smaller square in the cube. In the diagram above, points are midpoints of the edges of the inscribed cube. Therefore point , a vertex of the smaller cube, is also the center of the sphere. Point lies on the circumference of the sphere, so . is also the hypotenuse of right triangle . Similarly, is the hypotenuse of right triangle . If we let , then, by the properties of a right triangle, .
Using the Pythagorean Theorem, we can now solve for :
Since the side of the inscribed cube is , the volume is .
Example Question #221 : Psat Mathematics
If a cube has a surface area of , what is the difference between the volume of the cube and the surface area of the cube?
If the surface area is , then the area of one face must be . Therefore, the length of one edge must be This means that the volume of the cube is . We can now solve with:
Example Question #1 : Solid Geometry
What is the surface area of a cube with a volume of 1728 in3?
144 in2
864 in2
72 in2
1728 in2
432 in2
864 in2
This problem is relatively simple. We know that the volume of a cube is equal to s3, where s is the length of a given side of the cube. Therefore, to find our dimensions, we merely have to solve s3 = 1728. Taking the cubed root, we get s = 12.
Since the sides of a cube are all the same, the surface area of the cube is equal to 6 times the area of one face. For our dimensions, one face has an area of 12 * 12 or 144 in2. Therefore, the total surface area is 6 * 144 = 864 in2.
Example Question #1 : How To Find The Surface Area Of A Cube
A room has dimensions of 18ft by 15ft by 9ft. The last dimension is the height of the room. It has one door that is 3ft by 7ft and two windows, each 2ft by 5ft. There is no trim to the floor, wall, doors, or windows. What is the total exposed wall space?
2389ft2
1093ft2
1134ft2
553 ft2
594 ft2
553 ft2
If broken down into parts, this is an easy problem. It is first necessary to isolate the dimensions of the walls. If the room is 9 ft high, we know 18 x 15 designates the area of the floor and ceiling. Based on this, we know that the room has the following dimensions for the walls: 18 x 9 and 15 x 9. Since there are two of each, we can calculate the total area of walls - ignoring doors and windows - by doubling the sum of these two areas:
2 * (18 * 9 + 15 * 9) = 2 * (162 + 135) = 2 * 297 = 594 ft2
Now, we merely need to calculate the area "taken out" of the walls:
For the door: 3 * 7 = 21 ft2
For the windows: 2 * (2 * 5) = 20 ft2
The total wall space is therefore: 594 – 21 – 20 = 553 ft2
Example Question #1 : How To Find The Surface Area Of A Cube
A room has dimensions of 23ft by 17ft by 10ft. The last dimension is the height of the room. It has one door that is 2.5ft by 8ft and one window, 3ft by 6ft. There is no trim to the floor, wall, doors, or windows. If one can of paint covers 57 ft2 of surface area. How many cans of paint must be bought to paint the walls of the room.
15
13
11
18
14
14
If broken down into parts, this is an easy problem. It is first necessary to isolate the dimensions of the walls. If the room is 10ft high, we know 23 x 17 designates the area of the floor and ceiling. Based on this, we know that the room has the following dimensions for the walls: 23 x 10 and 17 x 10. Since there are two of each, we can calculate the total area of walls - ignoring doors and windows - by doubling the sum of these two areas:
2 * (23 * 10 + 17 * 10) = 2 * (230 + 170) = 2 * 400 = 800 ft2
Now, we merely need to calculate the area "taken out" of the walls:
For the door: 2.5 * 8 = 20 ft2
For the windows: 3 * 6 = 18 ft2
The total wall space is therefore: 800 – 20 – 18 = 762 ft2
Now, if one can of paint covers 57 ft2, we calculate the number of cans necessary by dividing the total exposed area by 57: 762/57 = (approx.) 13.37.
Since we cannot buy partial cans, we must purchase 14 cans.
Example Question #1 : How To Find The Surface Area Of A Cube
A certain cube has a side length of 25 m. How many square tiles, each with an area of 5 m2, are needed to fully cover the surface of the cube?
100
750
200
500
1000
750
A cube with a side length of 25m has a surface area of:
25m * 25m * 6 = 3,750 m2
(The surface area of a cube is equal to the area of one face of the cube multiplied by 6 sides. In other words, if the side of a cube is s, then the surface area of the cube is 6s2.)
Each square tile has an area of 5 m2.
Therefore, the total number of square tiles needed to fully cover the surface of the cube is:
3,750m2/5m2 = 750
Note: the volume of a cube with side length s is equal to s3. Therefore, if asked how many mini-cubes with side length n are needed to fill the original cube, the answer would be:
s3/n3