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) = 12(1.03)n

Part 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? (4 points)

Part B: What does the y-intercept of the graph of the function f(n) represent? (2 points)

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

Answer:

Step-by-step explanation:

A) 16.13 = 12(1.03)ⁿ

   1.344 = (1.03)ⁿ

ln(1.344) = n(ln(1.03))

           n = (ln(1.344)) / ln(1.03)

           n = 10.0062...

so a domain of 0 ≤ n ≤ 10 is reasonable

B) y-intercept is the height of the plant on day 0 which would be 12 cm

C) f(3) = 12(1.03)³ = 13.11 cm

average rate of change = Δh/Δn = (16.13 - 13.11) / (10 - 3) = 3.02 / 7

= 0.43 cm/day  or average growth per day

Answer 2

A) The reasonable domain to plot the growth function is given as; 0 ≤ n ≤ 10

B) The y-intercept of the graph of the function f(n) represents; the height of the plant after 0 days.

C) The average rate of change of the function f(n) from n = 3 to n = 10 is; 0.43 cm/day which means that the height increases by 0.43 cm each day.

A) We are given the function;

f(n) = 12 × (1.03)ⁿ

Where;

f(n) is the height of the plant after n days.

f(n) is given as 16.13. Thus;

16.13 = 12 × (1.03)ⁿ

(1.03)ⁿ = 16.13/12

(1.03)ⁿ = 1.3442

nIn 1.03 = In 1.3442

n = (In 1.3442)/(In 1.03)

n ≈ 10

Since the maximum number of days is 10, then the domain is: 0 ≤ n ≤ 10

B) The y-intercept will be the point at which n = 0. Thus at n = 0, we have;

f(0) = 12 × (1.03)^(0)

f(0) = 12 cm

C) The average rate of change from n = 3 to n = 10 is;

Δf(n) = (f(10) - f(3))/(10 - 3)

f(3) = 12 × (1.03)³

f(3) = 13.11

Thus;

Δf(n) = (16.13 - 13.11)/7

Δf(n) = 0.43 cm/day

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