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Find the volume of each prism.
SOLUTION: The volume V of a prism is V = Bh , where B is the area of a base and h is the height of the prism. The volume is 108 cm^3. ANSWER: 108 cm^3
SOLUTION: The volume V of a prism is V = Bh , where B is the area of a base and h is the height of the prism. ANSWER: 396 in^3
- the oblique rectangular prism shown SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. ANSWER: 26.95 m^3
- an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. ANSWER: 218.4 cm^3
Find the volume of each cylinder. Round to the nearest tenth.
SOLUTION: ANSWER: 206.4 ft^3
SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right cylinder and an oblique one of the same height and cross sectional area are same. ANSWER: 1357.2 m^3
- a cylinder with a diameter of 16 centimeters and a height of 5.1 centimeters SOLUTION: ANSWER: 1025.4 cm^3
- a cylinder with a radius of 4.2 inches and a height of 7.4 inches SOLUTION: ANSWER: 410.1 in^3
SOLUTION:
The base is a right triangle with a leg length of 9 cm and the hypotenuse of length 15 cm. Use the Pythagorean Theorem to find the height of the base. The height of the prism is 6 cm. ANSWER: 324 cm^3
SOLUTION:
If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. The volume V of a prism is V = Bh , where B is the area of a base and h is the height of the prism. B = 11.4 ft^2 and h = 5.1 ft. Therefore, the volume is ANSWER: 58.14 ft^3
- an oblique hexagonal prism with a height of 15 centimeters and with a base area of 136 square centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. ANSWER: 2040 cm^3
- a square prism with a base edge of 9.5 inches and a height of 17 inches SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. ANSWER: 1534.25 in^3 STRUCTURE Find the volume of each cylinder. Round to the nearest tenth.
- SOLUTION: r = 5 yd and h = 18 yd ANSWER: 1413.7 yd^3
SOLUTION:
r = 6 cm and h = 3.6 cm. ANSWER: 407.2 cm^3
SOLUTION: r = 5.5 in. Use the Pythagorean Theorem to find the height of the cylinder. Now you can find the volume. ANSWER: 823.0 in^3
b. When the radius is tripled, r = 3 r. So, when the radius is tripled, the volume is multiplied by 9. c. When the height and the radius are tripled, r = 3 r and h = 3 h. When the height and the radius are tripled, the volume is multiplied by 27. d. When the dimensions are exchanged, r = 8 and h = 5 cm. Compare to the original volume. The volume is multiplied by. ANSWER: a. The volume is multiplied by 3. b. The volume is multiplied by 3^2 or 9. c. The volume is multiplied by 3^3 or 27. d. The volume is multiplied by
23. INSULATION The insulated cup holds 16 ounces
of liquid. Find the volume of the insulating material,
rounded to the nearest cubic inch. SOLUTION: The volume of the insulated material is the difference between the volumes of the interior cylinder (which holds the liquid) and the entire cylinder (cup). The inner cylinder has a volume of 16 ounces (which converts to 28.875 cubic inches). Use this to find the radius of the inner cylinder. Note that the height of the inner cylinder is , due to the extra 0.5 insulation at the bottom of the cup. Therefore, the radius of the inner cylinder is inches, making the entire cup's radius to be inches. Find the volume of the entire cup. The volume of the insulating material is the difference between the volume of the inner cylinder and the volume of the entire cup.
Therefore, the volume of the insulated material is about 31 in^3. ANSWER: 31 in^3
24. MODELING The base of a rectangular paint tray is
sloped as shown below. Find the volume of paint it
takes to fill the tray.
SOLUTION:
The paint tray is a combination of a rectangular prism and a trapezoidal prism. The base of the rectangular prism is 8.9 cm by 45 cm and the height is 8.4 cm. The bases of the trapezoidal prism are 8.4 cm and 1. cm and the height of the base is
. The height of the trapezoidal prism is 45 cm. The total volume of the tray is the sum of the volumes of the two prisms. ANSWER:
14,735 cm³
25. CHANGING DIMENSIONS A cereal company
wants to increase the volume of each rectangular
prism container by 25% without changing the base.
Find the height of the new container if the original
had a base of 8 inches by 2 inches and a height of
12 inches. What would the height be if the surface
area of the container increased by 25%?
SOLUTION:
Find the volume of the original container. The volume of the new container is 125% of the original container, with the same base dimensions. Use 1.25 V and B to find h. Next, find the surface area of the original container. The surface area of the new container is 125% of the original container, with the same base dimensions. Use 1.25 S to find h.
inches
ANSWER:
15 in; 15.4 in.
SOLUTION:
The solid is a combination of a rectangular prism and a half cylinder.
The volume of this combination shape is 713.1 yd³.
ANSWER:
713.1 in^3
- FOOD A cylindrical can of baked potato chips has a height of 27 centimeters and a radius of 4 centimeters. A new can is advertised as being 30% larger than the regular can. If both cans have the same radius, what is the height of the larger can? SOLUTION: The volume of the smaller can is The volume of the new can is 130% of the smaller can, with the same radius. The height of the new can will be 35.1 cm. ANSWER: 35.1 cm
Find each measure to the nearest tenth.
- A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 centimeters. What is the height? SOLUTION: ANSWER: 5.7 cm
- A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume? SOLUTION: Use the surface area formula to solve for r. The radius is 6. Find the volume. ANSWER: 678.6 in^3
SOLUTION:
The circular bases at the top and bottom of the net indicate that this is a cylinder. If the middle piece were a rectangle, then the prism would be right. However, since the middle piece is a parallelogram, it is oblique. The radius is 1.8 m, the height is 4.8 m, and the slant height is 6 m. If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. ANSWER: 48.9 m^3
- SOIL A soil scientist wants to determine the bulk density of a potting soil to assess how well a specific plant will grow in it. The density of the soil sample is the ratio of its weight to its volume. a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0. pound per square inch is desirable for root growth? Explain. c. If a bag of this soil holds 2.5 cubic feet, what is its weight in pounds? SOLUTION: a. First calculate the volume of soil in the pot. Then divide the weight of the soil by the volume. The weight of the soil is the weight of the pot with soil minus the weight of the pot. W = 20 – 5 = 15 lbs. The soil density is thus: b. 0.0018 lb/in^3 is close to 0.0019 lb/in^3 so the plant should grow fairly well. c. ANSWER: a. 0.0019 lb / in^3
b. The plant should grow well in this soil since the bulk density of 0.0019 lb / in^3 is close to the desired bulk density of 0.0018 lb / in^3. c. 8.3 lb
- DESIGN Sketch and label (in inches) three different designs for a dry ingredient measuring cup that holds 1 cup. Be sure to include the dimensions in each drawing. (1 cup ≈ 14.4375 in^3 ) SOLUTION: For any cylindrical container, we have the following equation for volume: The last equation gives us a relation between the radius and height of the cylinder that must be fulfilled to get the desired volume. First, choose a suitable radius, say 1.85 in, and solve for the height. If we choose a height of say 4 in., then we can solve for the radius. For any rectangular container, the volume equation is: Choose numbers for any two of the dimensions and we can solve for the third. Let = 2.25 in. and w = 2.5 in. ANSWER: Sample answers:
the extra sand. If each trip takes him 30 seconds (or half a minute), this would take him , or 52 minutes, to provide her with the maximum amount of sand that she needs. c. The bucket is a cylinder. Ryann’s building site had no sand to start with. Jack goes at the same pace for every trip and doesn’t take any breaks. Each bucket is filled to the top and not overflowing with sand (or the average bucket-full equals the full capacity of the bucket). Jack uses both buckets at the same time. Ryann never stops to help Jack. Ryann needs enough sand to fill the entire volume. ANSWER: a. 52 min. b. The volume of the bucket is 256π or about 804 in³. If Jack carries two buckets at a time, then he can carry 1608 in³. The maximum volume of the castle is 165,888 in³. Therefore, it would take Jack 104 trips, or 52 minutes, to provide her with the maximum amount of sand that she needs. c. The bucket is a cylinder. Ryann’s building site had no sand to start with. Jack goes at the same pace for every trip and doesn’t take any breaks. Each bucket is filled to the top and not overflowing with sand (or the average bucket-full equals the full capacity of the bucket). Jack uses both buckets at the same time. Ryann never stops to help Jack. Ryann needs enough sand to fill the entire volume.
- Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism. SOLUTION: The base of the prism can be divided into 5 congruent triangles of a base 8 cm and the corresponding height 5.5 cm. So, the pentagonal prism is a combination of 5 triangular prisms of height 10 cm. Find the base area of each triangular prism. Therefore, the volume of the pentagonal prism is ANSWER: 1100 cm^3 ; Each triangular prism has a base area of or 22 cm^2 and a height of 10 cm.
- PATIOS Mr. Thomas is planning to remove an old patio and install a new rectangular concrete patio 20 feet long, 12 feet wide, and 4 inches thick. One contractor bid $2225 for the project. A second contractor bid $500 per cubic yard for the new patio and $700 for removal of the old patio. Which is the less expensive option? Explain. SOLUTION: Convert all of the dimensions to yards. 20 feet = yd 12 feet = 4 yd 4 in. = yd Find the volume. The total cost for the second contractor is about . Therefore, the second contractor is a less expensive option. ANSWER: Because 2.96 yd^3 of concrete are needed, the second contractor is less expensive at $2181.50.
- MULTIPLE REPRESENTATIONS In this problem, you will investigate right and oblique cylinders. a. GEOMETRIC Draw a right cylinder and an oblique cylinder with a height of 10 meters and a diameter of 6 meters. b. VERBAL A square prism has a height of 10 meters and a base edge of 6 meters. Is its volume greater than, less than, or equal to the volume of the cylinder? Explain. c. ANALYTICAL Describe which change affects the volume of the cylinder more: multiplying the height by x or multiplying the radius by x. Explain.
SOLUTION:
a. The oblique cylinder should look like the right cylinder (same height and size), except that it is pushed a little to the side, like a slinky. b. Find the volume of each. The volume of the square prism is greater. c. Do each scenario. Assuming x > 1, multiplying the radius by x makes the volume x^2 times greater. For example, if x = 0.5, then x^2 = 0.25, which is less than x. ANSWER: a.
3 by 4 by 10π ANSWER: Sample answers: a. 3 by 5 by 4 π b. 5 by 5 by
c. base with legs measuring 3 in. and 4 in., height
10π in.
- WRITING IN MATH Write a helpful response to the following question posted on an Internet gardening forum. I am new to gardening. The nursery will deliver a truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of soil? How do I know what to order? SOLUTION: The nursery means a cubic yard, which is 3^3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed. ANSWER: Sample answer: The nursery means a cubic yard, which is 3^3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.
- OPEN-ENDED Draw and label a prism that has a volume of 50 cubic centimeters. SOLUTION: Choose 3 values that multiply to make 50. The factors of 50 are 2, 5, 5, so these are the simplest values to choose. ANSWER: Sample answer:
- CONSTRUCT ARGUMENTS Determine whether the following statement is true or false. Explain. Two cylinders with the same height and the same lateral area must have the same volume. SOLUTION: The statement " Two cylinders with the same height and the same lateral area must have the same volume. " is true. If two cylinders have the same height ( h 1 = h 2 ) and the same lateral area ( L 1 = L 2 ), the circular bases must have the same area. The radii must also be equal. ANSWER: True; if two cylinders have the same height and the same lateral area, the circular bases must have the same area. Therefore, π r^2 h is the same for each cylinder.
- WRITING IN MATH How are the volume formulas for prisms and cylinders similar? How are they different? SOLUTION: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The base of a cylinder is a circle, so its area is πr^2. ANSWER: Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The base of a cylinder is a circle, so its area is πr^2.
- The rectangular prism shown here has a square base and a volume of 132.3 cubic inches. What is the perimeter of the base? A 30 in. B 17.64 in. C 16.8 in. D 4.2 in. SOLUTION: Substitute the given information into the formula for a rectangular prism and solve for x , one side of the square base. If the length of one side of the square base is 4. inches, then the perimeter is 4(4.2) or 16.8 inches. The correct choice is C. ANSWER: C
