## Engage NY Eureka Math Grade 6 Module 5 Lesson 12 Answer Key

### Eureka Math Grade 6 Module 5 Lesson 12 Example Answer Key

Example 1

a. Write a numerical expression for the volume of each of the rectangular prisms above.
(15 in,) (1$$\frac{1}{2}$$ in.) (3 in.)
(15 in.) (1 $$\frac{1}{2}$$ in.) (6 in.)
(15 in.) (1$$\frac{1}{2}$$ in.) (9 in.)

b. What do all of these expressions have in common? What do they represent?
All of the expressions have (15 in.) (1$$\frac{1}{2}$$ in.). This is the area of the base.

c. Rewrite the numerical expressions to show what they have in common.
(22$$\frac{1}{2}$$ in2)(3 in.) (22$$\frac{1}{2}$$ in2) (6 in.) (22$$\frac{1}{2}$$ in2) (9 in.)

d. If we know volume for a rectangular prism as length times width times height, what is another formula for volume that we could use based on these examples?
We could use (area of the base) (height), or area of the base times height.

e. What is the area of the base for all of the rectangular prisms?
(15 in.) (1$$\frac{1}{2}$$ in.) = 22 $$\frac{1}{2}$$ in2

f. Determine the volume of each rectangular prism using either method.
(15 in.)(1$$\frac{1}{2}$$ in.)(3 in.) = 67$$\frac{1}{2}$$ in3 or (22$$\frac{1}{4}$$ in2) (3 in.) = 67$$\frac{1}{2}$$ in3

(15 in.)(1$$\frac{1}{2}$$ in.)(6 in.) = 135 in3 or (22$$\frac{1}{2}$$ in2)(6 in.)= 135 in3

(15 in.)(1$$\frac{1}{2}$$ in.)(9 in.) = 202$$\frac{1}{2}$$ in3 or (22$$\frac{1}{2}$$ in2)(9 in.) = 204$$\frac{1}{2}$$ in3

g. How do the volumes of the first and second rectangular prisms compare? The volumes of the first and third?
135 in3 = 67 in3 × 2
202$$\frac{1}{2}$$ in3 = 67$$\frac{1}{2}$$ in3 × 3

The volume of the second prism is twice that of the first because the height is doubled. The volume of the third prism is three times as much as the first because the height is triple the first prism’s height.

Example 2:
The base of a rectangular prism has an area of 3$$\frac{1}{4}$$ in2. The height of the prism is 2$$\frac{1}{2}$$ in. Determine the volume of the rectangular prism.
V = Area of base × height
V = (3$$\frac{1}{4}$$ in2) (2$$\frac{1}{2}$$ in.)
V = ($$\frac{13}{4}$$ in2) ($$\frac{5}{2}$$in.)
V = $$\frac{65}{8}$$ in3

Extension:

Question 1.
A company is creating a rectangular prism that must have a volume of 6 ft3. The company also knows that the area of the base must be 2$$\frac{1}{2}$$ ft2. How can you use what you learned today about volume to determine the height of the rectangular prism?
I know that the volume can be calculated by multiplying the area of the base times the height. So, if I needed the height instead, I would do the opposite. I would divide the volume by the area of the base to determine the height.
V = Area of base × height
6ft3 = (2$$\frac{1}{2}$$ ft2)h
6 ft3 ÷ 2$$\frac{1}{2}$$ ft2 = h
2$$\frac{2}{5}$$ ft. = h

### Eureka Math Grade 6 Module 5 Lesson 12 Problem Set Answer Key

Question 1.
Determine the volume of the rectangular prism.

V = l w h
V = (1$$\frac{1}{2}$$ m) ($$\frac{1}{2}$$ m) ($$\frac{7}{8}$$ m)
V = $$\frac{21}{32}$$ m3

Question 2.
The area of the base of a rectangular prism is 4ft2, and the height is 2 ft. Determine the volume of the rectangular prism.

Question 3.
The length of a rectangular prism is 3$$\frac{1}{2}$$ times as long as the width. The height is $$\frac{1}{4}$$ of the width. The width is 3 cm. Determine the volume.
Width = 3cm
Length = 3 cm × 3$$\frac{1}{2}$$ = $$\frac{21}{2}$$ cm
Height = 3 cm × $$\frac{1}{4}$$ = $$\frac{3}{4}$$ cm
V = l w h
V = ($$\frac{21}{2}$$ cm) (3 cm) ($$\frac{3}{4}$$ cm)
V = $$\frac{189}{8}$$ cm3

Question 4.

a. Write numerical expressions to represent the volume in two different ways, and explain what each reveals.
(10$$\frac{1}{2}$$ in.) (1$$\frac{2}{3}$$ in.) (6 in.) represents the product of three edge lengths. ($$\frac{35}{2}$$ in2) (6 in.) represents the product of the base area times the height. Answers will vary.

b. Determine the volume of the rectangular prism.
(10$$\frac{1}{2}$$ in.)(1$$\frac{2}{3}$$ in.)(6 in.)= 105 in3 or
($$\frac{35}{2}$$ in2) (6 in.)= 105 in3

Question 5.
An aquarium in the shape of a rectangular prism has the following dimensions: length = 50 cm, width = 25$$\frac{1}{2}$$ cm, and height = 30 cm.

a. Write numerical expressions to represent the volume in two different ways, and explain what each reveals.
(50 cm) (25$$\frac{1}{2}$$ cm) (30$$\frac{1}{2}$$ cm) represents the product of the three edge lengths. (1,275 cm2) (30$$\frac{1}{2}$$ cm) represents the base area times the height.

b. Determine the volume of the rectangular prism.
(1.275 cm2) (30$$\frac{1}{2}$$ cm) = 38,887$$\frac{1}{2}$$ cm3

Question 6.
The area of the base in this rectangular prism is fixed at 36 cm2. As the height of the rectangular prism changes, the volume will also change as a result.

a. Complete the table of values to determine the various heights and volumes.

b. Write an equation to represent the relationship in the table. Be sure to define the variables used In the equation.
Let X be the height of the rectangular prism in centimeters.
Let y be the volume of the rectangular prism in cubic centimeters.
36x = y

c. What is the unit rate for this proportional relationship? What does it mean in this situation?
The unit rate is 36.
For every centimeter of height, the volume increases by 36 cm3 because the area of the base is 36 cm2. In order to determine the volume, multiply the height by 36.

Question 7.
The volume of a rectangular prism is 16.328 cm3. The height is 3. 14 cm.

a. Let B represent the area of the base of the rectangular prism. Write an equation that relates the volume, the area of the base, and the height.
16.328 = 3.14B

b. Solve the equation for B.
$$\frac{16.328}{3.14}=\frac{3.14 B}{3.14}$$
B = 5.2
The area of the base is 5.2 cm2.

### Eureka Math Grade 6 Module 5 Lesson 12 Exit Ticket Answer Key

Question 1.
Determine the volume of the rectangular prism in two different ways.

V = l. w. h
V = ($$\frac{3}{4}$$ ft.) ($$\frac{3}{8}$$ ft.) ($$\frac{3}{4}$$ ft.)
V = $$\frac{27}{128}$$ ft3

V = Area of base. height
V = ($$\frac{9}{32}$$ ft2) . ($$\frac{3}{4}$$ ft2.)
V = $$\frac{27}{128}$$ ft3

Question 2.
The area of the base of a rectangular prism is 12 cm2, and the height is 3 cm. Determine the volume of the rectangular prism.
V = (12 cm2) (3$$\frac{1}{3}$$ cm)
V = $$\frac{120}{3}$$ cm3