WP7a
Practice - Volumes and Surface Areas
TRY TO ANSWER IN TWO AND A HALF MINUTES.
QUESTIONS
QUESTIONS
1. What is the surface area of a cylinder which has a diameter of 28" and a height of 66"?
2. If the surface area of a sphere is 144 π, what is the radius of the sphere?
3. The volume of a cylinder is 48 π cubic inches. If the height of the cylinder is 12 inches, what is the diameter of the circular end?
4. A cabinet whose corners are all 90 degrees has the following dimensions: 24 inches by 72 inches by 18 inches. What is the volume of the cabinet, in square feet?
5. A cone has slant height of 6" and a diameter of 4". What is the surface area of the cone?
6. The volume of a cube is 32 units. What is the volume of a smaller cube for which each side is 1/2 that of the first cube?
7. A box measures 24" by 72" by 18". What is the volume of the box in cubic feet?
8. A room has these dimensions: 9' high x 12' wide x 15' long. A gallon of paint covers 350 square feet. How many gallons are needed to paint the four walls and the ceiling?
9. You want to cover a sphere with gold leaf. You have one sheet of gold leaf which is 10" by 10". What is the radius of the largest sphere you can cover with the sheet?
10. A sphere weighs ten pounds. What would a sphere with twice the diameter of the first one weigh?
11. A cylinder has diameter = 8 and height = 12. What is the volume of the cylinder?
12. What is the surface area of a cylinder whose radius is 2 and whose height is 5?
13. You have a cylinder 8' in diameter and 12' high. Water flows into the cylinder at a rate of 600 cubic feet/hour, beginning at 11 am. At what time does the tank start to overflow? (Use 3.14 for π).
14. You have an aquarium which is 30" long and 20" high. You want to fill it with sand to a depth of 6". How many bags of sand will it take if each bag contains 1 cubic foot?
ANSWERS
1. From the built-in formula sheet with the test, you can see that the formula for the surface area of a cylinder is 2 π(rh + r²). Since the diameter is 28", the radius is 14". So the answer is 2 π(14x66 + 14² ) = 2 π (924 + 196), or 2 π (1120), or 2240 π.
2. From the built-in formula sheet, you can see that the surface area of a sphere is 4 π r² , where r is the radius. Therefore 144 π = 4 π r² or 4r² = 144, r ² = 36, so r = 6.
3. From the built-in formula sheet, you find that the volume of a cylinder is π r²h. So 48 = r² h. If h = 12, r² = 48/12, or 4. So r = 2 and the diameter is equal to twice the radius, or four inches.
4. First convert each of the dimensions from inches to feet: 2' by 6' x 1½' . The volume is the product of these four dimensions, or 12 x 1½ = 18 square feet.
5. From the formula sheet built into the test, you can see that the surface area of a cone is πrs + πr² . The radius r is half the diameter. Plugging r = 2 and s = 6 into this formula gives (12 + 4 )π = 16 π .
6. Let s = the length of the side of the first cube. The volume of a cube is s³ . (This formula isn't on the drop-down sheet; you'll have to remember it.) So s³ =32. The area of the second cube is (s/2)³ = s³ x (1/2)³ = s³/8. Since the volume of the first cube is x³ ,the volume of the second cube is 1/8 that of the first cube.
7. Convert all of the measurements from inches to feet: 2' x 6' x 1½'. The volume is the product of the three numbers, which is 2 x 6 x 1½ = 18 cubic feet. The volume formula is not on the drop-down sheet.
8. Each long wall is 15' long by 9' high, with area = 135 square feet. Each short wall is 12' long by 9' high, with area = 108 square feet. The ceiling is 15' by 12', with area = 180 square feet. The total area to be painted is twice each wall areas plus the ceiling area, which is 2(125) + 2(108) + 180 = 250 + 216 + 180 = 646 square foot. 646/350 = 1.85, so you will need two gallons to paint the room.
9. The sheet has 10 x 10 = 100 square inches. The formula for the surface area of a sphere is 4πr² (from the drop-down formula sheet). So 100 = 4πr² . Dividing each side by four gives 25 = πr². Taking the square root of each side gives r = 5√π.
10. Weight is proportional to volume. The formula for the volume of a sphere (from the drop-down sheet) is (4/3)πr³. So the volume of the larger sphere would be (4/3)π(4r)³ . Since 4³ = 64, the volume of the second sphere would be (4/3)π64r³ , so the volume and the weight of the larger sphere would be 64 times that of the smaller sphere.
11. The formula for the volume of a cylinder (from the drop-down sheet) is V = πr²h. Since the diameter = 8, the radius r =4. Plugging r = 4 and h = 12 into the formula, you get V = π x 4² x h = (16 x 12)π = 192π .
12. The formula for the surface area of a cylinder (from the drop-down sheet) is SA = 2πrh + 2πr². Since r = 2 and h = 5, SA = 2 π (2x5) + 2π x 2² = 20π + 8 π = 28π.
13. The formula for the volume of a cylinder (from the drop-down sheet) is V = πr²h. Since the diameter = 8, the radius r = 4. Plugging these two numbers into the formula gives V = π x 4² x 12 = 3.14 x 16 x 12 = 602.88 cubic feet. At a rate of flow of 600 cubic feet/hour, the tank will overflow just after noon.
14. One cubic foot of sand equals 12 x 12 x 12 = 1728 cubic inches of sand. The aquarium's bottom has area 30 x 20 = 600 square inches. So if the desired height is 6 inches, then the resulting volume is 6 x 600, which is equal to 1728 x, where x is the number of bags of sand. 3600/1728 = 2.08 bags of sand.