When an original amount is reduced by a consistent rate over a period of time, exponential decay is occurring. The purpose of this page is to use exponential decay, an exponential function, to predict outcomes. Answers and Explanations for the worksheet, Exponential Functions and a Hurricane can be found here.
Here’s an exponential decay function:
y = a(1-b)x
- y: Final amount remaining after the decay over a period of time
- a: The original amount
- x: Time
- The decay factor is (1-b).
- The variable, b, is percent decrease in decimal form.
Three Paths to Percent Decrease (b)
- The percent decrease is mentioned in the story.
- The percent decrease is expressed in a function.
- The percent decrease is hidden in a set of data.
Exponential Decay in Real Life: Hurricane Grendel
On Thursday, meteorologists confirm that Hurricane Grendel will hit Houston in 5 days. Gasoline, water, and plywood dwindle. As the days pass, fewer people will remain in the Houston Metropolitan area.
Use the following information to complete the exercises.
Cases of Water at Houston Super Center
The manager of Houston Super Center uses this function to project the number of cases of water at the store:
y = 550,000(1-.25)x, where y represents the number of cases of water remaining x days after the hurricane announcement is made.
Gasoline at Chev-aco
The owner of Chev-aco predicts that the number of gallons of gasoline will have a percent decrease of 12.5% each day after Thursday.
Plywood at Home Hardware
Home Hardware counted 6,400 square feet of plywood on hand before Thursday, the day that the storm was announced. The vice president of lumber projects the daily supply of plywood.
- Thursday: 6,400 square feet
- Friday: 3,200 square feet
- Saturday: 1,600 square feet
- Sunday : 800 square feet
People in the Houston Metropolitan Area
Thursday, 5,600,000 people are in the city. Here's the mayor's projection of the number of people remaining in the Houston area:
- Thursday: 5,600,000 people
- Friday: 1,400,000 people
- Saturday: 350,000 people
- Sunday: 87,500 people
Use the information above to complete the following exercises.
Answers and explanations for Exponential Functions and a Hurricane are included.
1. WATER
A. Write a function to predict how many cases of water will remain at Houston Super Mart on Monday.
Answer: y = 550,000(1-.25)4
Explanation: The manager of Houston Super Center uses this function to project the number of cases of water at the store:
y = 550,000(1-.25)x, where y represents the number of cases of water remaining x days after the hurricane announcement is made.
Since the announcement was made on Thursday, Monday is the 4th day after Hurricane Grendel is announced. Therefore, x = 4
B. How many cases of water remain at Houston Super Mart on Monday?
Answer: 174,023.4375 cases of water
Explanation: 550,000(1-.25)4 = 174,023.4375
2. PLYWOOD
A. What is the percent decrease for the amount of plywood?
Answer: 50%
Explanation:
A. Choose the data for 2 consecutive days: Thursday: 6,400 square feet; Friday: 3,200 square feet.
B. Percent decrease = (older- newer)/older
(6400 - 3200)/6400 = 50%
C. Verify that 50% is correct by testing 2 other consecutive days: Saturday: 1,600 square feet; Sunday: 800 square feetD. Percent decrease = (older- newer)/older
(1600-800)/1600 = 50%
B. What is the decay factor for the amount of plywood?
Answer: .50
Explanation: Decay factor = (1-b) = (1-.50) = .50
C. What is the amount of plywood at the store on Thursday?
Answer: 6,400 square feet
D. Write a function that will help you predict the amount of plywood that Home Hardware will have on Monday.
Answer: y = 6400(1-.50)4
Explanation:
Fill in the blanks of the exponential decay function: y = a(1-b)x
a = 6400, b = .50, x = 4
E. How much plywood will Home Hardware have on Monday?
Answer: 400 square feet
Explanation: 6400(1-.50)4 = 400
3. HOUSTON POPULATION
A. What is the percent decrease in the number people who remain in Houston after the Hurricane is announced?
Answer: 75%
Explanation:
A. Choose the data for 2 consecutive days: Thursday: 5,600,000 people; Friday: 1,400,000 people.
B. Percent decrease = (older- newer)/older
(5,600,000 - 1,400,000)/5,600,000 = 75%
C. Verify that 75% is correct by testing 2 other consecutive days: Saturday: 350,000 people; Sunday: 87,500 peopleD. Percent decrease = (older- newer)/older
(350,000-87,500)/350,000 = 75%
B. What is the decay factor for the number of people remaining in Houston?
Answer: .25
Explanation: Decay factor = (1-b) = (1-.75) = .25
C. What is the population on Thursday, the day the storm is announced?
Answer: 5,600,000
Explanation: Thursday, 5,600,000 people are in the city.
D. Write a formula to determine how many people will remain in Houston on Monday.
Answer:
Explanation:
Answer: y = 5600000(1-.75)4
Explanation:
Fill in the blanks of the exponential decay function: y = a(1-b)x
a = 5600000, b = .75, x = 4
E. Use the formula to determine how many people will remain in Houston on Monday.
Answer: 21,875 people
Explanation: 5600000(1-.75)4 = 21875