1. The box shown below is 5 units long, 3 units wide, and 4 units high. How many unit cubes will fit in the box ? What is the volume of the box ?
2. Find the volume of the right prism shown below.
3. Find the volume of the right cylinder shown below.
4. The volume of the cube shown below is 100 ft^{3}. Find the value of x.
5. The volume of the right cylinder shown below is 4561 m^{3}. Find the value of x.
6. If a concrete weighs 145 pounds per cubic foot, find the weight of the concrete block shown below.
1. Answer :
The base of the box is 5 units by 3 units. This means 5 • 3 or 15 unit cubes, will cover the base.
Solution (a) :
Three more layers of 15 cubes each can be placed on top of the lower layer to fill the box. Because the box contains 4 layers with 15 cubes in each layer, the box contains a total of 4 • 15, or 60 unit cubes.
Solution (b) :
Because the box is completely filled by the 60 cubes and each cube has a volume of 1 cubic unit, it follows that the volume of the box is 60 • 1, or 60 cubic units.
2. Answer :
The area of the base is
B = 1/2 ⋅ (3)(4)
B = 6 cm^{2}
The height is
h = 2 cm
Formula for volume of a right prism is
V = Bh
Substitute 6 for B and 2 for h.
V = (6)(2)
V = 12
So, the volume of the right prism is 12 cubic cm.
3. Answer :
Formula for volume of a right cylinder is
V = πr^{2}h
Substitute 8 for r and 6 for h.
V = π(8^{2})(6)
Simplify.
V = 384π
Use calculator.
V ≈ 1206.37
So, the volume of the right cylinder is about 1206.37 cubic inches.
4. Answer :
A side length of the cube is x feet.
Formula for volume of a cube :
V = s^{3}
Substitute 100 for V and x for s.
100 = x^{3}
Take cube root on both sides.
∛100 = ∛x
4.64 ≈ x
So, the value of x is about 4.64
5. Answer :
Formula for volume of a right cylinder is
V = πr^{2}h
Substitute 4561 for V, x for r and 12 for h.
4561 = πx^{3}(12)
4561 = 12πx^{3}
Divide each side by 12π.
4561/12π = x^{2}
Find the positive square root.
11 ≈ x
So, the value of x is about 11.
6. Answer :
To find the weight of the concrete block shown, we need to find its volume.
The area of the base can be found as follows :
B = Area larger rectangle - 2 ⋅ Area of small rectangle
B = (1.31)(0.66) - 2(0.33)(0.39)
B ≈ 0.61 ft^{2}
Using the formula for the volume of a prism, the volume is
V = Bh
V ≈ 0.61(0.66)
V ≈ 0.40 ft^{3}
To find the weight of the block, multiply the pounds per cubic foot, 145 lb/ft^{3}, by the number of cubic feet, 0.40 ft^{3}.
Weight = [145 lb/ft^{3}] ⋅ [0.40 ft^{3}]
Simplify.
Weight ≈ 58 lb
So, the weight of the concrete block is about 58 pounds.
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