Question

A scientist is studying the growth of a particular species of plant. He writes the following equation to show the height of the plant f(n), in cm, after n days:
f(n) = 10(1.02)n

Part A: When the scientist concluded his study, the height of the plant was approximately 11.04 cm. What is a reasonable domain to plot the growth function?

Part B: What does the y-intercept of the graph of the function f(n) represent?

Part C: What is the average rate of change of the function f(n) from n = 1 to n = 5, and what does it represent?

Answer 1

The average rate of change of the function from n = 1 to n = 5 is 3.36 plants per day

The reasonable domain

The function is given as:

f(n) = 10(1.02)^n

The maximum height is 11.04

So, we have:

10(1.02)^n = 11.04

Divide by 10

1.02^n = 1.104

Take the log of both sides

n * log(1.02) = log(1.104)

Divide by log(1.02)

n = 5

Hence, the reasonable domain is 0 ≤ n < 5

The y-intercept

We have:

f(n) = 10(1.02)^n

Set n to 0

f(0) = 10(1.02)^0

Evaluate

f(0) = 10

Hence, the y-intercept is 10

The average rate of change

We have:

f(n) = 10(1.02)^n

Calculate f(1) and f(5)

f(1) = 10(1.02)^1 = 10.2

f(5) = 10(1.02)^5 = 10.2

The average rate is then calculated as:

m = (f(5) - f(1))/(5 - 1)

This gives

m =(11.04 - 10.2)(5 - 1)

Evaluate

m = 3.36

Hence, the average rate of change of the function f(n) from n = 1 to n = 5 is 3.36

Read more about exponential functions at:

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