How to find surface area - HSPT Math

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Question

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?

Answer

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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Question

An upright cylinder with a height of 30 and a radius of 5 is in a big tub being filled with oil. If only the top 10% of the cylinder is visible, what is the surface area of the submerged cylinder?

Answer

The height of the submerged part of the cylinder is 27cm. 2πrh + πr2 is equal to 270π + 25π = 295π

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Question

The diameter of the lid of a right cylindrical soup can is 5 in. If the can is 12 inches tall and the label costs $0.00125 per square inch to print, what is the cost to produce a label for a can? (Round to the nearest cent.)

Answer

The general mechanics of this problem are simple. The lateral area of a right cylinder (excluding its top and bottom) is equal to the circumference of the top times the height of the cylinder. Therefore, the area of this can's surface is: 5π * 12 or 60π. If the cost per square inch is $0.00125, a single label will cost 0.00125 * 60π or $0.075π or approximately $0.24.

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Question

Aluminum is sold to a soup manufacturer at a rate of $0.0015 per square inch. The cans are made so that the ends perfectly fit on the cylindrical body of the can. It costs $0.00125 to attach the ends to the can. The outer label (not covering the top / bottom) costs $0.0001 per in2 to print and stick to the can. The label must be 2 inches longer than circumference of the can. Ignoring any potential waste, what is the manufacturing cost (to the nearest cent) for a can with a radius of 5 inches and a height of 12 inches?

Answer

We have the following categories to consider:

= ( + ) * 0.0015

= () * 0.0001

= 2 * 0.00125 = $0.0025

The area of ends of the can are each equal to π52 or 25π. For two ends, that is 50π.

The lateral area of the can is equal to the circumference of the top times the height, or 2 π * r * h = 2 * 5 * 12 * π = 120π.

Therefore, the total surface area of the aluminum can is 120π + 50π = 170π. The cost is 170π * 0.0015 = 0.255π, or approximately $0.80.

The area of the label is NOT the same as the lateral area of the can. (Recall that it must be 2 inches longer than the circumference of the can.) Therefore, the area of the label is (2 + 2 * π * 5) * 12 = (2 + 10π) * 12 = 24 + 120π. Multiply this by 0.0001 to get 0.0024 + 0.012π = (approximately) $0.04.

Therefore, the total cost is approximately 0.80 + 0.04 + 0.0025 = $0.8425, or $0.84.

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Question

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?

Answer

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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Question

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.

Answer

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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Question

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?

Answer

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 6_s_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 _s_3. 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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Question

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?

Answer

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)πr_2_h

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

486_π_ = (1/3)πr_2 2_r* = (2/3)_πr_3

Multiply both sides by (3/2_π_): 729 = _r_3

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 * _s_2, or for our data, A = 6 * 1112 = 73,926 in2

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Question

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?

Answer

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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Question

According to regulations, the maximum radius of a (spherical) bowling ball is 7.11 centimeters. What does that make the maximum surface area, to the nearest square centimeter?

(Note: disregard the holes)

Answer

Set and use the formula for the surface area of a sphere:

$A = 4\pi r^{2} = 4\pi\cdot 7.11^{2} \approx 635; cm^{2 }$

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Question

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?

Answer

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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Question

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

Answer

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 :

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

square centimeters

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Question

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.

Answer

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

.

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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Question

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 square feet. How many cans of the paint will the manager need to buy?

Answer

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:

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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Question

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

Answer

The surface area of the cylinder can be calculated by setting and 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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Question

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

Answer

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 .

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Question

A rectangular prism has a volume of 70 m3. If the length, width, and height of the prism are integers measured in meters, which of the following is NOT a possible measure of the surface area of the prism measured in square meters?

Answer

Since the volume is the product of length, width, and height, and each of these three dimensions are integers, it is important to know the factors of the volume. 70 = (2)(5)(7). This implies that each of these factors (and only these factors with the exception of 1) will show up in the three dimensions exactly once. This creates precisely the following five possibilities:

2, 5, 7

SA = 2((2)(5)+(2)(7)+(5)(7)) = 118

1, 7, 10

SA = 2((1)(7)+(1)(10)+(7)(10)) = 174

1, 5, 14

SA = 2((1)(5)+(1)(14)+(5)(14)) = 178

1, 2, 35

SA = 2((1)(2)+(1)(35)+(2)(35)) = 214

1, 1, 70

SA = 2((1)(1)+(1)(70)+(1)(70)) = 282

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Question

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

Answer

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

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Question

The three sides of a rectangular box all have integer unit lengths. If each of the side lengths is greater than one unit, and if the volume of the box is 182 cubic units, what is the surface area of the box in square units?

Answer

Let's call the side lengths of the box l, w, and h. We are told that l, w, and h must all be integer lengths greater than one. We are also told that the volume of the box is 182 cubic units.

Since the volume of a rectangular box is the product of its side lengths, this means that lwh must equal 182.

(l)(w)(h) = 182.

In order to determine possible values of l, w, and h, it would help us to figure out the factors of 182. We want to express 182 as a product of three integers each greater than 1.

Let's factor 182. Because 182 is even, it is divisible by 2.

182 = 2(91).

91 is equal to the product of 7 and 13.

Thus, 182 = 2(7)(13).

This means that the lengths of the box must be 2, 7, and 13 units.

In order to find the surface area, we can use the following formula:

surface area = 2lw + 2lh + 2hw.

surface area = 2(2)(7) + 2(2)(13) + 2(7)(13)

= 28 + 52 + 182

= 262 square units.

The answer is 262.

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Question

A solid sphere is cut in half to form two solid hemispheres. What is the ratio of the surface area of one of the hemispheres to the surface area of the entire sphere before it was cut?

Answer

The surface area of the sphere before it was cut is equal to the following:

surface area of solid sphere = 4_πr_2, where r is the length of the radius.

Each hemisphere will have the following shape:

In order to determine the surface area of the hemisphere, we must find the surface area of the flat region and the curved region. The flat region will have a surface area equal to the area of a circle with radius r.

area of flat part of hemisphere = _πr_2

The surface area of the curved portion of the hemisphere will equal one-half of the surface area of the uncut sphere, which we established to be 4_πr_2.

area of curved part of hemisphere = (1/2)4_πr_2 = 2_πr_2

The total surface area of the hemisphere will be equal to the sum of the surface areas of the flat part and curved part of the hemisphere.

total surface area of hemisphere = _πr_2 + 2_πr_2 = 3_πr_2

Finally, we must find the ratio of the surface area of the hemisphere to the surface area of the uncut sphere.

ratio = (3_πr_2)/(4_πr_2) = 3/4

The answer is 3/4.

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