**Problem 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 ?

**Problem 2 :**

Find the volume of the right prism shown below.

**Problem 3 :**

Find the volume of the right cylinder shown below.

**Problem 4 : **

The volume of the cube shown below is 100 ft^{3}. Find the value of x.

**Problem 5 : **

The volume of the right cylinder shown below is 4561 m^{3}. Find the value of x.

**Problem 6 :**

If a concrete weighs 145 pounds per cubic foot, find the weight of the concrete block shown below.

**Problem 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 ?

**Solution : **

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.

**Problem 2 :**

Find the volume of the right prism shown below.

**Solution : **

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.

**Problem 3 :**

Find the volume of the right cylinder shown below.

**Solution : **

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.

**Problem 4 : **

The volume of the cube shown below is 100 ft^{3}. Find the value of x.

**Solution : **

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

**Problem 5 : **

The volume of the right cylinder shown below is 4561 m^{3}. Find the value of x.

**Solution : **

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.

**Problem 6 :**

If a concrete weighs 145 pounds per cubic foot, find the weight of the concrete block shown below.

**Solution : **

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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