COMMON CORE MATHEMATICS CURRICULUM
7•6
Lesson 27
Lesson 27: Real-World Volume Problems
Student Outcomes

Students use the volume formula for a right prism (𝑉 = 𝐵ℎ) to solve volume problems involving rate of flow.
Lesson Notes
Students apply their knowledge of volume to real-world contexts, specifically problems involving rate of flow of liquid.
These problems are not unlike problems involving distance, speed, and time; instead of manipulating the formula
𝐷 = 𝑟𝑡, students work with the formula 𝑉 = 𝑟𝑡.
Classwork
Scaffolding:
Opening (6 minutes)

Imagine a car is traveling at 50 mph. How far will it go in 30 minutes?

It travels 50 miles in one hour; therefore, it travels 25 miles in 30
minutes.

You just made use of the formula 𝐷 = 𝑟𝑡 to solve that problem. Today we will
use a similar formula.

Here is a sample of the real-world context that we will be studying today.
Imagine a faucet turned onto the maximum level flows at a rate of 1 gal. in 25
seconds. How long will it take to fill a 10-gallon tank at this rate?

What are the different quantities in this question?




𝟎. 𝟓
𝟐𝟓
𝟏
𝟓𝟎
𝟏. 𝟓
𝟕𝟓
𝟐
𝟏𝟎𝟎
𝟐. 𝟓
𝟏𝟐𝟓
Seconds
Gal.
𝟐𝟓
𝟏
𝟓𝟎
𝟐
Water that flows at a rate (volume per unit of time) for a given amount of time yields
a volume.
𝟕𝟓
𝟑
𝟏𝟎𝟎
𝟒
𝑉 = 𝑟𝑡
𝟏𝟐𝟓
𝟓
𝟏𝟓𝟎
𝟔
𝟏𝟕𝟓
𝟕
𝟐𝟎𝟎
𝟖
𝟐𝟐𝟓
𝟗
𝟐𝟓𝟎
𝟏𝟎
How can we tackle this problem?


.
25
What is the relationship between the quantities?


The constant is
1
Miles
 Remind students how to
use a ratio table by posing
questions such as the
following: How far will the
car travel in half an hour?
In two hours?
Create a ratio table for this situation. What is the constant?


Rate, time, and volume
Hours
𝑉
𝑟
=𝑡
25 𝑠𝑒𝑐.
(10 𝑔𝑎𝑙. )
= 250 sec., or 4 min. and 10 sec.
1 𝑔𝑎𝑙.
We will use the formula 𝑉 = 𝑟𝑡 in other contexts that involve a rate of flow.
Lesson 27:
Date:
Real-World Volume Problems
3/21/16
285
Lesson 27
COMMON CORE MATHEMATICS CURRICULUM
7•6
Example 1 (8 minutes)
Scaffolding:
Example 1
ft3
A swimming pool holds 𝟏𝟎, 𝟎𝟎𝟎 of water when filled. Jon and Anne want to fill the pool with a
garden hose. The garden hose can fill a five-gallon bucket in 𝟑𝟎 seconds. If each cubic foot is
about 𝟕. 𝟓 gallons, find the flow rate of the garden hose in gallons per minute and in cubic feet per
minute. About how long will it take to fill the pool with a garden hose? If the hose is turned on
Monday morning at 𝟖 a.m., approximately when will the pool be filled?

If the hose fills a 5-gallon bucket in 30 seconds, how much would it fill in 1
minute? Find the flow rate in gallons per minute.



𝟏
𝟑
take to fill the pool with a garden
hose? If the hose is turned on Monday
morning at 𝟖 a.m., approximately
when will the pool be filled?
It would fill 10 gallons in 1 min.; therefore, the flow rate is 10 gal./min.
3
1 𝑓𝑡
1
= 1 𝑓𝑡 3
7.5 𝑔𝑎𝑙.
3

(10 𝑔𝑎𝑙. )

Therefore, the flow rate of the garden hose in cubic feet per minute is 1 ft3/min.
1
3
How many minutes would it take to fill the 10,000 ft3 pool?
10,000 𝑓𝑡 3
= 7,500 min.
1 𝑓𝑡3
1 (
)
3 1 𝑚𝑖𝑛.
How many days and hours is 7,500 minutes?


A swimming pool holds 𝟏𝟎, 𝟎𝟎𝟎 ft3 of
water when filled. Jon and Anne want
to fill the pool with a garden hose. The
flow rate of the garden hose is
𝟏 ft3/min. About how long will it
Find the flow rate in cubic feet per minute.


 To complete the problem
without the step involving
conversion of units, use
the following problem:
(7,500 𝑚𝑖𝑛. )
1 ℎ𝑟.
= 125 h, or 5 days and 5 hours.
60 𝑚𝑖𝑛.
At what time will the pool be filled?

The pool begins to fill at 8 a.m. on Monday, so 5 days and 5 hours later on Saturday at 1 p.m., the pool
will be filled.
Example 2 (8 minutes)
Example 2
A square pipe (a rectangular prism shaped pipe) with inside dimensions of 𝟐 in. × 𝟐 in. has water flowing through it at
the speed of 𝟑 ft./s. The water flows into a pool in the shape of a right triangular prism, with a base in the shape of a
right isosceles triangle and with legs that are each 𝟓 feet in length. How long will it take for the water to reach a depth of
𝟒 feet?

Let students begin the problem on their own. Depending on their progress, the teacher may want to share the
following once they are done. If students are struggling, share the information up front.

Since the water is traveling at 3 ft./s., every second the volume of water flowing out of the pipe is the same as
the volume of a right rectangular prism with dimensions 2 in. × 2 in. × 3 ft.

The volume of this prism in cubic feet is ft. ×
pipe every second is
1 3
ft .
12
Lesson 27:
Date:
1
6
1
1 3
ft. × 3 ft.=
ft ; and the volume of water flowing out of the
6
12
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Lesson 27
COMMON CORE MATHEMATICS CURRICULUM

What is the volume of water that will flow in one minute?


1
𝑓𝑡 3
12
1 𝑠𝑒𝑐.
∙
60 𝑠𝑒𝑐.
1 𝑚𝑖𝑛.
= 5 ft3/min.
What is the volume of water that will be in the pool once the water reaches a depth of four feet?


7•6
1
2
The volume of water in the pool will be (5 ft.)(5 ft.)(4 ft.) = 50 ft3.
How long will it take for the pool to fill?

50 𝑓𝑡 3
5
𝑓𝑡3
𝑚𝑖𝑛.
= 10 min.; it will take 10 min. to fill the pool.
Exercise 1 (8 minutes)
Students will have to find volumes of two composite right rectangular prisms in this exercise. Remind students as they
work on finding the volume of the lower level of the fountain that the volume of the whole top level must be subtracted
from the inner volume of the lower level. This does not however require the whole height of the top level; the relevant
height for the volume that must be subtracted is 2 ft. (see calculation in solution).
Exercise 1
A park fountain is about to be turned on in the spring after having been off all winter long. The fountain flows out of the
top level and into the bottom level until both are full, at which point the water is just recycled from top to bottom
through an internal pipe. The outer wall of the top level, a right square prism, is five feet in length, the thickness of the
stone between outer and inner wall is 𝟏 ft., and the depth is 𝟏 ft. The bottom level, also a right square prism, has an
outer wall that is 𝟏𝟏 ft. long with a 𝟐-ft. thickness between the outer and inner wall and a depth of 𝟐 ft. Water flows
through a 𝟑 in. × 𝟑 in. square pipe into the top level of the fountain at a rate of 𝟒 ft./s. Approximately how long will it
take for both levels of the fountain to fill completely?
Volume of top:
𝟑 ft. × 𝟑 ft. × 𝟏 ft. = 𝟗 ft3
Volume of bottom:
(𝟕 ft. × 𝟕 ft. × 𝟐 ft. ) − (𝟓 ft. × 𝟓 ft. × 𝟐 ft.)
= 𝟒𝟖 ft3
Combined volume of both levels:
𝟗 ft3 + 𝟒𝟖 ft3 = 𝟓𝟕 ft3
A flow of 𝟒 ft./s. is equal to a volume of a right rectangular prism with dimensions of 𝟑 in. × 𝟑 in. × 𝟒 ft. The volume of
this prism is
𝟏
𝟒
ft. ×
𝟏
𝟏
𝟏
ft. × 𝟒 ft.= ft3; ft3 of water flows every second.
𝟒
𝟒
𝟒
Volume of water that will flow in one minute:
𝟏 𝟑
𝒇𝒕 𝟔𝟎 𝒔𝒆𝒄.
𝒇𝒕𝟑
𝟒
∙
= 𝟏𝟓
𝟏 𝒔𝒆𝒄. 𝟏 𝒎𝒊𝒏.
𝒎𝒊𝒏.
Time needed to fill both fountain levels:
𝟓𝟕 𝒇𝒕𝟑
𝟏𝟓
Lesson 27:
Date:
𝒇𝒕𝟑
𝒎𝒊𝒏.
= 𝟑. 𝟖 min.; it will take 𝟑. 𝟖 min. to fill both fountain levels.
Real-World Volume Problems
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COMMON CORE MATHEMATICS CURRICULUM
Lesson 27
7•6
Exercise 2 (7 minutes)
Exercise 2
A decorative bathroom faucet has a 𝟑 in. × 𝟑 in. square pipe that flows into a basin in the shape of an isosceles trapezoid
prism like the one shown in the diagram. If it takes one minute and twenty seconds to fill the basin completely, what is
the approximate rate of flow from the faucet in feet per second?
Volume of the basin in cubic inches:
𝟏
𝟐
(𝟑 in. + 𝟏𝟓 in.)(𝟏𝟎 in.) × 𝟒. 𝟓 in. = 𝟒𝟎𝟓 in3
Approximate volume of the basin in cubic feet:
𝟑
(𝟒𝟎𝟓 in3 ) (
𝟏 𝒇𝒕
𝟏,𝟕𝟐𝟖 𝒊𝒏
𝟑)
= 𝟎. 𝟐𝟑𝟒𝟑𝟕𝟓 ft3
Based on the rate of water flowing out the faucet, the volume of water can also be calculated as follows:
𝟏
𝟒
ft. ∙
𝟏
𝟒
ft. ∙ 𝒔 ft. = 𝟎. 𝟐𝟑𝟒𝟑𝟕𝟓 ft3
Therefore, the rate of flow of water is 𝒔 = 𝟑. 𝟕𝟓 ft./s.
Closing (1 minute)

What does it mean for water to flow through a square pipe?


The pipe can be visualized as a right rectangular prism.
If a 2 in. × 2 in. square pipe has a rate of flow of 4 ft./ s., what is the volume of water that flows from the pipe
every second?

1
6
1
1
6
9
ft. ∙ ft. ∙ 4 ft. = ft3
Exit Ticket (7 minutes)
Lesson 27:
Date:
Real-World Volume Problems
3/21/16
288
Lesson 27
COMMON CORE MATHEMATICS CURRICULUM
Name
7•6
Date
Lesson 27: Real-World Volume Problems
Exit Ticket
Jim wants to know how much his family spends on water for showers. Water costs $1.50 for 1,000 gallons. His family
averages 4 showers per day. The average length of a shower is 10 minutes. He places a bucket in his shower and turns
on the water. After one minute, the bucket has 2.5 gallons of water. About how much money does his family spend on
water for showers in a 30-day month?
Lesson 27:
Date:
Real-World Volume Problems
3/21/16
289
Lesson 27
COMMON CORE MATHEMATICS CURRICULUM
7•6
Exit Ticket Sample Solutions
Jim wants to know how much his family spends on water for showers. Water costs $𝟏. 𝟓𝟎 for 𝟏, 𝟎𝟎𝟎 gallons. His family
averages 𝟒 showers per day. The average length of a shower is 𝟏𝟎 minutes. He places a bucket in his shower and turns
on the water. After one minute, the bucket has 𝟐. 𝟓 gallons of water. About how much money does his family spend on
water for showers in a 𝟑𝟎-day month?
𝟐.𝟓 𝒈𝒂𝒍.
Number of gallons of water in one day of showering (four, ten-minute showers): 𝟒(𝟏𝟎 min.) (
Number of gallons of water in 𝟑𝟎 days: (𝟑𝟎 days) (
Cost of showering for 𝟑𝟎 days: (𝟑, 𝟎𝟎𝟎 gal.) (
𝟏 𝒎𝒊𝒏.
) = 𝟏𝟎𝟎 gal.
𝟏𝟎𝟎 𝒈𝒂𝒍.
) = 𝟑, 𝟎𝟎𝟎 gal.
𝟏 𝒅𝒂𝒚
$𝟏.𝟓𝟎
)
𝟏,𝟎𝟎𝟎 𝒈𝒂𝒍.
= $𝟒. 𝟓𝟎
The family spends $𝟒. 𝟓𝟎 in a 𝟑𝟎-day month on water for showers.
Problem Set Sample Solutions
1.
Harvey puts a container in the shape of a right rectangular prism under a spot in the roof that is leaking. Rain water
is dripping into the container at an average rate of 𝟏𝟐 drops a minute. The container Harvey places under the leak
has a length and width of 𝟓 cm and a height of 𝟏𝟎 cm. Assuming each raindrop is roughly 𝟏 cm3, approximately how
long does Harvey have before the container overflows?
Volume of the container in cubic centimeters:
𝟓 cm × 𝟓 cm × 𝟏𝟎 cm = 𝟐𝟓𝟎 cm3
Number of minutes until the container is filled with rainwater:
(𝟐𝟓𝟎 cm3 ) (
2.
𝟏 𝒎𝒊𝒏.
)
𝟏𝟐 𝒄𝒎𝟑
≈ 𝟐𝟎. 𝟖 min.
A large square pipe has inside dimensions 𝟑 in. × 𝟑 in., and a small square pipe has inside dimensions 𝟏 in. × 𝟏 in.
Water travels through each of the pipes at the same constant speed. If the large pipe can fill a pool in 𝟐 hours, how
long will it take the small pipe to fill the same pool?
𝟏 𝟏
If 𝒔 is the speed of the water in feet per minute, then in one minute the large pipe provides ∙ ∙ 𝒔 ft3. In one minute
the small pipe provides one-ninth as much,
𝟏
∙
𝟏
𝟏𝟐 𝟏𝟐
𝟒 𝟒
∙ 𝒔 ft3. Therefore, it will take the small pipe nine times as long. It
will take the small pipe 𝟏𝟖 hours to fill the pool.
3.
A pool contains 𝟏𝟐, 𝟎𝟎𝟎 ft3 of water and needs to be drained. At 𝟖 a.m., a pump is turned on that drains water at
the rate of 𝟏𝟎 ft3 per minute. Two hours later, at 𝟏𝟎 a.m., a second pump is activated that drains water at the rate
of 𝟖 ft3 per minute. At what time will the pool be empty?
Water drained in the first two hours:
𝟏𝟎 𝒇𝒕𝟑
𝟏 𝒎𝒊𝒏.
∙ 𝟏𝟐𝟎 min. = 𝟏, 𝟐𝟎𝟎 ft3
Volume of water that still needs to be drained: 𝟏𝟐, 𝟎𝟎𝟎 ft3 − 𝟏, 𝟐𝟎𝟎 ft3 = 𝟏𝟎, 𝟖𝟎𝟎 ft3
Amount of time needed to drain remaining water with both pumps working is
𝟏 𝒎𝒊𝒏.
𝟏𝟎, 𝟖𝟎𝟎 ft3 (
𝟏𝟖 𝒇𝒕𝟑
) = 𝟔𝟎𝟎 min. or 𝟏𝟎 h.
Total time needed to drain the pool is 𝟏𝟐 h., so the pool will drain completely at 𝟖 p.m.
Lesson 27:
Date:
Real-World Volume Problems
3/21/16
290
Lesson 27
COMMON CORE MATHEMATICS CURRICULUM
4.
7•6
In the previous problem, if water starts flowing into the pool at noon at the rate of 𝟑 ft3 per minute, how much
longer will it take to drain the pool?
At noon, the first pump will have been on for four hours, and the second pump will have been on for two hours. The
cubic feet of water drained by the two pumps together at noon is
𝟑
𝟑
𝟏𝟎 𝒇𝒕
𝟖 𝒇𝒕
) + 𝟏𝟐𝟎 min.(
)
𝟏 𝒎𝒊𝒏.
𝟏 𝒎𝒊𝒏.
𝟐𝟒𝟎 min.(
= 𝟑, 𝟑𝟔𝟎 ft3
Volume of water that still needs to be drained:
𝟏𝟐, 𝟎𝟎𝟎 ft3 − 𝟑, 𝟑𝟔𝟎 ft3 = 𝟖, 𝟔𝟒𝟎 ft3
If water is entering the pool at 𝟑 ft3, but leaving it at 𝟏𝟖 ft3, the net effect is that water is leaving the pool at 𝟏𝟓 ft3.
Amount of time needed to drain remaining water with both pumps working and water flowing in is
𝟏 𝒎𝒊𝒏.
𝟖, 𝟔𝟒𝟎 ft3 (
𝟏𝟓 𝒇𝒕
𝟑)
= 𝟓𝟕𝟔 min., or 𝟗 h and 𝟑𝟔 min. The pool will finish draining at 𝟗: 𝟑𝟔 p.m. the same day. It will
take an additional 𝟏 hour and 𝟑𝟔 minutes to drain the pool.
5.
A pool contains 𝟔, 𝟎𝟎𝟎 ft3 of water. Pump A can drain the pool in 𝟏𝟓 hours, pump B can drain it in 12 hours, and
pump C can drain it in 𝟏𝟎 hours. How long will it take all three pumps working together to drain the pool?
Rate at which Pump A drains the pool:
Rate at which Pump B drains the pool:
Rate at which Pump C drains the pool:
Together the pumps drain the pool at (
𝟏
𝟏𝟓
𝟏
𝟏𝟐
𝟏
𝟏𝟎
pool per hour
pool per hour
pool per hour
𝟏
𝟏
𝟏
𝟏
+ + ) pool per hour or pool per hour.
𝟏𝟓 𝟏𝟐 𝟏𝟎
𝟒
Therefore, it will take 𝟒 hours
to drain the pool when all three pumps are working together.
6.
A 𝟐, 𝟎𝟎𝟎 gallon fish aquarium can be filled by water flowing at a constant rate in 𝟏𝟎 hours. When a decorative rock
is placed in the aquarium, it can be filled in 𝟗. 𝟓 hours. Find the volume of the rock in cubic feet (𝟏 ft3 = 𝟕. 𝟒𝟖 gal.).
Rate of water flow into aquarium:
𝟐, 𝟎𝟎𝟎 𝒈𝒂𝒍. 𝟐𝟎𝟎 𝒈𝒂𝒍.
=
𝟏𝟎 𝒉𝒓𝒔.
𝟏 𝒉𝒓.
Since it takes half an hour less time to fill the aquarium with the rock inside, the volume of the rock is
𝟐𝟎𝟎 𝒈𝒂𝒍.
𝟏 𝒉𝒓.
(𝟎. 𝟓 hr.) = 𝟏𝟎𝟎 gal.
Volume of the rock:
𝟏𝟎𝟎 gal.(
𝟏 𝒇𝒕𝟑
𝟕.𝟒𝟖 𝒈𝒂𝒍.
) ≈ 𝟏𝟑. 𝟑𝟕 ft3; the volume of the rock is approximately 𝟏𝟑. 𝟑𝟕 ft3.
Lesson 27:
Date:
Real-World Volume Problems
3/21/16
291