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HSPT Math : HSPT Mathematics

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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?

Possible Answers:

594 ft²

1134ft²

2389ft²

1093ft²

553 ft²

Correct answer:

553 ft²

Explanation:

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

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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.

Possible Answers:

Correct answer:

Explanation:

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.

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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?

Possible Answers:

200

1000

100

750

500

Correct answer:

750

Explanation:

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 6s^2.)

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

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Example Question #11 : Cubes

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:

73,926 in²

69,984 in²

513.375 in²

486 in²

726 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)πr²* 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

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Example Question #3 : Non Cubic Prisms

Angie is painting a 2 foot cube for a play she is in. She needs $25\hspace{1 mm}mL$ of paint for every square foot she paints. How much paint does she need?

Possible Answers:

$600\hspace{1 mm}mL$

$100\hspace{1 mm}mL$

None of the available answers

$1.041\overline{6}\hspace{1 mm}mL$

It is impossible to convert between metric units and feet.

Correct answer:

$600\hspace{1 mm}mL$

Explanation:

First we must calculate the surface area of the cube. We know that there are six surfaces and each surface has the same area:

$Area=6(2^2)=6\times 4=24\hspace{1 mm}feet^2$

Now we will determine the amount of paint needed

$24\hspace{1 mm}feet^2\times \frac{25\hspace{1 mm}mL}{1\hspace{1 mm}foot^2}=600\hspace{1 mm}mL$

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Example Question #1 : How To Find The Surface Area Of A Sphere

A spherical orange fits snugly inside a small cubical box such that each of the six walls of the box just barely touches the surface of the orange. If the volume of the box is 64 cubic inches, what is the surface area of the orange in square inches?

Possible Answers:

16π

128π

32π

256π

64π

Correct answer:

16π

Explanation:

The volume of a cube is found by V = s³. Since V = 64, s = 4. The side of the cube is the same as the diameter of the sphere. Since d = 4, r = 2. The surface area of a sphere is found by SA = 4π(r²) = 4π(2²) = 16π.

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Example Question #1 : How To Find The Area Of A Rectangle

Steve's bedroom measures 20' by 18' by 8' high. He wants to paint the ceiling and all four walls using a paint that gets 360 square feet of coverage per gallon. A one-gallon can of the paint Steve wants costs $36.00; a one-quart can costs $13.00. What is the least amount of money that Steve can expect to spend on paint in order to paint his room?

Possible Answers:

$\$98$

$\$111$

$\$85$

$\$108$

$\$121$

Correct answer:

$\$108$

Explanation:

Two of the walls have area $20 \cdot 8 = 160\;\textrm{in}^{2}$ ; two have area $18 \cdot 8 = 144\;\textrm{in}^{2}$ ; the ceiling has area $20 \cdot 18 = 360\;\textrm{in}^{2}$ .

Therefore, the total area Steve wants to cover is

$A = 160 + 160 + 144 + 144 + 360 =968 \; \textrm{in}^{2}$

Divide 968 by 360 to get the number of gallons Steve needs to paint his bedroom:

$968 \div 360 \approx 2.7$

Since $2 \frac{1}{2} < 2.7 < 2\frac{3}{4}$ , Steve needs to purchase either two gallon cans and three quart cans, or three gallon cans.

The first option will cost him $36 \cdot2 + 13 \cdot3 = \$111$ ; the second option will cost him $36 \cdot3 = \$108$ . The latter is the more economical option.

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Example Question #242 : Geometry

Prism

Give the surface area of the above box in square centimeters.

Possible Answers:

$28,800 \textrm { cm}^{2}$

$324,000 \textrm { cm}^{2}$

$216,000 \textrm { cm}^{2}$

$32,400 \textrm { cm}^{2}$

$21,600 \textrm { cm}^{2}$

Correct answer:

$28,800 \textrm { cm}^{2}$

Explanation:

100 centimeters make one meter, so convert each of the dimensions of the box by multiplying by 100.

$0.9 \times 100 = 90$ centimeters

$0.6 \times 100 = 60$ centimeters

Use the surface area formula, substituting L = 90, W = H = 60 :

A = 2LH + 2LW + 2HW

$A = 2 \cdot 90 \cdot 60 + 2\cdot 90 \cdot 60 + 2 \cdot 60 \cdot 60$

A = 10,800 + 10,800 + 7,200 = 28,800 square centimeters

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Example Question #243 : Geometry

Square

Note: Figure NOT drawn to scale.

Refer to the above diagram, which shows a square. Give the ratio of the area of the yellow region to that of the white region.

Possible Answers:

57:7

The correct answer is not given among the other choices.

55:9

7:1

27:5

Correct answer:

55:9

Explanation:

The area of the entire square is the square of the length of a side, or

$80 \times 80 = 6,400$ .

The area of the right triangle is half the product of its legs, or

$\frac{1}{2} \times 30 \times 60 = 900$ .

The area of the yellow region is therefore the difference of the two, or

6,400 - 900 = 5,500 .

The ratio of the area of the yellow region to that of the white region is

$\frac{5,500}{900} = \frac{5,500 \div 100}{900\div 100} = \frac{55}{9}$ ; that is, 55 to 9.

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Example Question #601 : Problem Solving

Swimming_pool

The above depicts a rectangular swimming pool for an apartment. The pool is five feet deep everywhere.

An apartment manager wants to paint the four sides and the bottom of the swimming pool. One one-gallon can of the paint he wants to use covers 350 square feet. How many cans of the paint will the manager need to buy?

Possible Answers:

Correct answer:

Explanation:

The bottom of the swimming pool has area

$50 \times 35 = 1,750$ square feet.

There are two sides whose area is

$50 \times 5 = 250$ square feet,

and two sides whose area is

$35 \times 5 = 175$ square feet.

Add the areas:

1,750 + 250+ 250+ 175+175= 2,600 square feet.

One one-gallon can of paint covers 350 square feet, so divide:

$2,600 \div 350 \approx 7.4$

Seven full gallons and part of another are required, so eight is the correct answer.

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HSPT Math : HSPT Mathematics

Study concepts, example questions & explanations for HSPT Math

All HSPT Math Resources

Example Questions

Example Question #1 : How To Find The Surface Area Of A Cube

Example Question #1 : How To Find The Surface Area Of A Cube

Example Question #1 : How To Find The Surface Area Of A Cube

Example Question #11 : Cubes

Example Question #3 : Non Cubic Prisms

Example Question #1 : How To Find The Surface Area Of A Sphere

Example Question #1 : How To Find The Area Of A Rectangle

Example Question #242 : Geometry

Example Question #243 : Geometry

Example Question #601 : Problem Solving

All HSPT Math Resources

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