Activity
A marine biologist estimates that a structure of coral has a volume of 1,200 cubic centimeters and that its volume doubles each year.
1. Write an equation of the form representing the relationship, where is time in years since the coral was measured and is volume of coral in cubic centimeters. (You need to figure out what numbers and are in this situation.)
Compare your answer:
2. Find the volume of the coral when is 5, 1, 0, –1, and –2. Note that when , this represents one year before the marine biologist made the original estimate.
Time (years) | 5 | 1 | 0 | -1 | -2 |
Volume of Coral (cubic centimeters) | a.____ | b.____ | c.____ | d.____ | e.____ |
a. Enter the volume of coral when t = 5. Do not enter commas in your answer.
3800 cubic centimeters
b. Enter the volume of coral when t = 1. Do not enter commas in your answer.
2400 cubic centimeters
c. Enter the volume of coral when t = 0. Do not enter commas in your answer.
1200 cubic centimeters
d. Enter the volume of coral when t = -1. Do not enter commas in your answer.
600 cubic centimeters
e. Enter the volume of coral when t = -2. Do not enter commas in your answer.
300 cubic centimeters
3. What does it mean, in this situation, when is –2?
Compare your answer:
This represents the time 2 years before the scientist made the estimate of 1,200 cubic centimeters.
4. In a certain year, the volume of the coral is 37.5 cubic centimeters. Which year is this? Be prepared to show your reasoning.
Compare your answer:
is –5, or 5 years before the scientist made the estimate of 1,200 cubic centimeters.
Self Check
Additional Resources
Interpreting Negative Exponents in Exponential Growth
Let’s look at a problem involving an exponential growth equation and negative exponents.
A forest fire has been burning for several days. The burned area, in acres, is given by the equation , where is the number of days since the area of the fire was first measured.
1. Let’s complete the table.
, days since first measurement | Process column | , acres burned since fire started |
0 | 4800 | |
–1 | 2400 | |
–2 | 0 | c. _____ |
–3 | 0 | d. _____ |
–5 | 0 | e. _____ |
To determine the value of , given , follow these steps:
Example 1
Determine the value for when
Step 1 -Substitute the given domain value for d into the expression.
Step 2 -Evaluate the exponential value. Recall that a negative exponent represents how many times the reciprocal of the base should be multiplied.
Step 3 -Evaluate the remaining expression to determine .
Example 2
Determine the value for when
Step 1 -Substitute the given domain value for d into the expression.
Step 2 -Evaluate the exponential value. Recall that a negative exponent represents how many times the reciprocal of the base should be multiplied.
Step 3 -Evaluate the remaining expression to determine .
Example 3
Determine the value for when
Step 1 -Substitute the given domain value for d into the expression.
Step 2 -Evaluate the exponential value. Recall that a negative exponent represents how many times the reciprocal of the base should be multiplied.
Step 3 -Evaluate the remaining expression to determine .
Example 4
What does the value of tell you about the area burned in the fire when ?
The value at represents the fact that 2,400 acres had burned 1 day before the first measurement was taken.
Example 5
How much area had the fire burned a week before it measured 4,800 acres? Explain your reasoning.
Since there are 7 days in a week, to find the area burned a week before the fire was measured, we need to determine the value of when .
Step 1 -Substitute the given domain value for d into the expression.
Step 2 -Evaluate the exponential value. Recall that a negative exponent represents how many times the reciprocal of the base should be multiplied.
Step 3 -Evaluate the remaining expression to determine .
acres had burned a week before the fire measured acres. The value is found by substituting for d since that represents days before the first measurement of acres.
Try it
Try It: Interpreting Negative Exponents in Exponential Growth
The value of a home in 2015 was $400,000. Its value has doubled each decade.
1. If is the value of the home, in dollars, write an equation for in terms of , the number of decades since 2015.
Compare your answers:
2. What is when ?
3. What does this value mean?
Compare your answers:
This means the home had a value of 50,000 in 1985, three decades before 2015.
4. What does this value mean?
Compare your answers:
This means the home had a value of 200,000 in 2005, one decade before 2015.
5. What is when ?