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HSPT Math : Problem Solving

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Example Questions

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:

750

500

100

200

1000

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 #242 : Geometry

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:

726 in²

69,984 in²

486 in²

513.375 in²

73,926 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 : How To Find The Surface Area Of A Prism

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:

$600\hspace{1 mm}mL$

It is impossible to convert between metric units and feet.

None of the available answers

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

$100\hspace{1 mm}mL$

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:

64π

128π

16π

256π

32π

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 #241 : Geometry

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:

$\$111$

$\$85$

$\$98$

$\$121$

$\$108$

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:

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

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

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

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

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

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 #101 : 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:

7:1

The correct answer is not given among the other choices.

27:5

55:9

57:7

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 #162 : Geometry

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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Example Question #2141 : Hspt Mathematics

Cylinder

The surface area of the above cylinder is 80% of what number?

Possible Answers:

$5,250 \pi$

$45,000 \pi$

$28,000 \pi$

$1,680 \pi$

Correct answer:

$5,250 \pi$

Explanation:

The surface area of the cylinder can be calculated by setting r = 30 and h = 40 in the formula

$A =2 \pi r (r + h)$

$A =2 \pi \cdot 30 (30 + 40)$

$A =2 \pi \cdot 30 \cdot 70$

$A = 4,200 \pi$

This is 80% of the number

$4,200 \pi \div \frac{80}{100} = 4,200 \pi \cdot \frac{100} {80}= 5,250 \pi$

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Example Question #601 : Problem Solving

What is the surface area of a cube with a side edge of 7?

Possible Answers:

168

294

196

343

Correct answer:

294

Explanation:

First you must find the area of one side of the cube which is side squared.

$\\A=s^2 \\A=7^2 \\A=49$ .

Then you multiply that area by six since that is how many sides a cube has.

Your answer is 49*6=294 .

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HSPT Math : Problem Solving

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 #242 : Geometry

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

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

Example Question #241 : Geometry

Example Question #242 : Geometry

Example Question #101 : Geometry

Example Question #162 : Geometry

Example Question #2141 : Hspt Mathematics

Example Question #601 : Problem Solving

All HSPT Math Resources

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