Lesson 27: Real-World Volume Problems
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- Julian Miles
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1 Student Outcomes Students use the volume formula for a right prism ( ) 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, students work with the formula. Classwork Opening (6 minutes) Imagine a car is traveling at mph. How far will it go in minutes? It travels miles in one hour; therefore, it travels miles in minutes. You just made use of the formula 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 gal. in seconds. How long will it take to fill a -gallon tank at this rate? What are the different quantities in this question? Rate, time, and volume Scaffolding: Hours 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? The constant is. What is the relationship between the quantities? Water that flows at a rate (volume per unit of time) for a given amount of time yields a volume. How can we tackle this problem? sec., or min. and sec. We will use the formula in other contexts that involve a rate of flow. Seconds Gal. Date: 4/9/14 285
2 Example 1 (8 minutes) Example 1 A swimming pool holds ft 3 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 seconds. If each cubic foot is about 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 a.m., approximately when will the pool be filled? If the hose fills a -gallon bucket in seconds, how much would it fill in minute? Find the flow rate in gallons per minute. It would fill gallons in min.; therefore, the flow rate is gal./min. Find the flow rate in cubic feet per minute. Scaffolding: To complete the problem without the step involving conversion of units, use the following problem: A swimming pool holds ft 3 of water when filled. Jon and Anne want to fill the pool with a garden hose. The flow rate of the garden hose is ft 3 /min. About how long will it take to fill the pool with a garden hose? If the hose is turned on Monday morning at a.m., approximately when will the pool be filled? Therefore, the flow rate of the garden hose in cubic feet per minute is ft 3 /min. How many minutes would it take to fill the ft 3 pool? min. How many days and hours is minutes? h, or days and hours. At what time will the pool be filled? The pool begins to fill at a.m. on Monday, so days and hours later on Saturday at 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 in. in. has water flowing through it at the speed of 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 feet in length. How long will it take for the water to reach a depth of 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 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 in. in. ft. The volume of this prism in cubic feet is ft. ft. ft. ft 3 ; and the volume of water flowing out of the pipe every second is ft 3. Date: 4/9/14 286
3 What is the volume of water that will flow in one minute? ft 3 /min. What is the volume of water that will be in the pool once the water reaches a depth of four feet? The volume of water in the pool will be ft. ft. ft. ft 3. How long will it take for the pool to fill? min.; it will take 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 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 ft., and the depth is ft. The bottom level, also a right square prism, has an outer wall that is ft. long with a -ft. thickness between the outer and inner wall and a depth of ft. Water flows through a in. in. square pipe into the top level of the fountain at a rate of ft./s. Approximately how long will it take for both levels of the fountain to fill completely? Volume of top: ft. ft. ft. ft 3 Volume of bottom: ft. ft. ft ft. ft. ft. ft 3 Combined volume of both levels: ft 3 ft 3 ft 3 A flow of ft./s. is equal to a volume of a right rectangular prism with dimensions of in. in. ft. The volume of this prism is ft. ft. ft. ft 3 ; ft 3 of water flows every second. Volume of water that will flow in one minute: Time needed to fill both fountain levels: min.; it will take min. to fill both fountain levels. Date: 4/9/14 287
4 Exercise 2 (7 minutes) Exercise 2 A decorative bathroom faucet has a in. 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: in. in. in. in. in 3 Approximate volume of the basin in cubic feet: in 3 ft 3 Based on the rate of water flowing out the faucet, the volume of water can also be calculated as follows: ft. ft. ft. ft 3 Therefore, the rate of flow of water is 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 in. in. square pipe has a rate of flow of ft./ s., what is the volume of water that flows from the pipe every second? ft. ft. ft. ft 3 Exit Ticket (7 minutes) Date: 4/9/14 288
5 Name Date Exit Ticket Jim wants to know how much his family spends on water for showers. Water costs for gallons. His family averages showers per day. The average length of a shower is minutes. He places a bucket in his shower and turns on the water. After one minute, the bucket has gallons of water. About how much money does his family spend on water for showers in a -day month? Date: 4/9/14 289
6 Exit Ticket Sample Solutions Jim wants to know how much his family spends on water for showers. Water costs for gallons. His family averages showers per day. The average length of a shower is minutes. He places a bucket in his shower and turns on the water. After one minute, the bucket has gallons of water. About how much money does his family spend on water for showers in a -day month? Number of gallons of water in one day of showering (four, ten-minute showers): min. gal. Number of gallons of water in days: days gal. Cost of showering for days: gal. The family spends in a -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 drops a minute. The container Harvey places under the leak has a length and width of cm and a height of cm Assuming each raindrop is roughly cm 3, approximately how long does Harvey have before the container overflows? Volume of the container in cubic centimeters: cm cm cm cm 3 Number of minutes until the container is filled with rainwater: cm 3 min. 2. A large square pipe has inside dimensions in. in., and a small square pipe has inside dimensions in. in. Water travels through each of the pipes at the same constant speed. If the large pipe can fill a pool in hours, how long will it take the small pipe to fill the same pool? If is the speed of the water in feet per minute, then in one minute the large pipe provides ft 3. In one minute the small pipe provides one-ninth as much, will take the small pipe hours to fill the pool. ft 3. Therefore, it will take the small pipe nine times as long. It 3. A pool contains ft 3 of water and needs to be drained. At a.m., a pump is turned on that drains water at the rate of ft 3 per minute. Two hours later, at a.m., a second pump is activated that drains water at the rate of ft 3 per minute. At what time will the pool be empty? Water drained in the first two hours: min. ft 3 Volume of water that still needs to be drained: ft 3 ft 3 ft 3 Amount of time needed to drain remaining water with both pumps working is ft 3 min. or h. Total time needed to drain the pool is h., so the pool will drain completely at p.m. Date: 4/9/14 290
7 4. In the previous problem, if water starts flowing into the pool at noon at the rate of ft 3 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 min. min. ft 3 Volume of water that still needs to be drained: ft 3 ft 3 ft 3 If water is entering the pool at ft 3, but leaving it at ft 3, the net effect is that water is leaving the pool at ft 3. Amount of time needed to drain remaining water with both pumps working and water flowing in is ft 3 min., or h and min. The pool will finish draining at p.m. the same day. It will take an additional hour and minutes to drain the pool. 5. A pool contains ft 3 of water. Pump A can drain the pool in hours, pump B can drain it in 12 hours, and pump C can drain it in 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: pool per hour pool per hour pool per hour Together the pumps drain the pool at pool per hour or pool per hour. Therefore, it will take hours to drain the pool when all three pumps are working together. 6. A gallon fish aquarium can be filled by water flowing at a constant rate in hours. When a decorative rock is placed in the aquarium, it can be filled in hours. Find the volume of the rock in cubic feet ( ft 3 gal.). Rate of water flow into aquarium: Since it takes half an hour less time to fill the aquarium with the rock inside, the volume of the rock is hr. gal. Volume of the rock: gal. ft 3 ; the volume of the rock is approximately ft 3. Date: 4/9/14 291
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