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 in f(n), in cm, after n days:f(n)=12(1.03)^nPart A: When the scientist concluded his study, the height of the plant was approximately 16.13 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=3 to n=10 and what does it represent?

Answer 1

The modeled equation by the scientist is

f(n) = 12(1.03)^n

Where f(n) is the height and n = days

The scientist concluded his study that he height of the plant was approximately 16.13cm

Therefore, f(n) = 16.13cm

From the modeled equation

f(n) = 12(1.03)^n

16.13 = 12 (1.03)^n

Divide both sides by 12

16.13/12 = 12(1.03)^n /12

16.13/12 = (1.03)^n

1.344 = (1.03)^n

Take the logarithms of both side

Log 1.344 = Log (1.03)^n

Log 1.344 = nlog(1.03)

Log 1.344 = 0.1284

Log 1.03 = 0.01284

Hence, the equation becomes

0.1284 = 0.01284n

Divide both sides by 0.01284

0.1284 / 0.01284 = 0.01284n/0.01284

n = 0.1284/0.01284

n = 10

Therefore, the reasonable domain to plot the growth function is n less than or equal to 10

`n\leq\text{ 10}`