Answer:
A) 0 ≤ n ≤ 7
B) initial height of plant = 8 cm
C) average rate of change = 0.475 cm/day.
Plant grew an average of 0.475 cm per day from day 2 to day 6
Step-by-step explanation:
[tex]f(n)=8(1.05)^n[/tex]
Part A
Given height = 11.26 cm
[tex]\implies 8(1.05)^n=11.26[/tex]
[tex]\implies(1.05)^n=\dfrac{11.26}{8}[/tex]
[tex]\implies(1.05)^n=1.4075[/tex]
Taking natural logs:
[tex]\implies \ln(1.05)^n=\ln(1.4075)[/tex]
[tex]\implies n\ln(1.05)=\ln(1.4075)[/tex]
[tex]\implies n=\dfrac{\ln(1.4075)}{\ln(1.05)}[/tex]
[tex]\implies n=7.005819448[/tex]
[tex]\implies n \approx7[/tex]
Therefore, the scientist concluded his study after 7 days, so the domain is 0 ≤ n ≤ 7
Part B
y-intercept is when the curve intersects the y-axis (when n = 0):
[tex]\implies f(0)=8(1.05)&^0=8\times1=8[/tex]
The y-intercept represents the initial height of the plant, which is 8 cm
Part C
Rate of change = change in y ÷ change in x
⇒ rate of change = change in f(n) ÷ change in n
[tex]\implies f(2)=8(1.05)^2=8.82[/tex]
[tex]\implies f(6)=8(1.05)^6=10.72076513...[/tex]
[tex]\implies \textsf{rate of change} \ =\dfrac{f(6)-f(2)}{6-2}=0.4751912813...[/tex]
So the average rate of change was 0.475 cm/day. This indicates that the plant grew an average of 0.475 cm per day from day 2 to day 6 (over a period of 4 days).
ETA: please see attached graph. The gray dotted curve is the function, and the black solid curve is the function with the restricted domain of 0≤n≤7 since the study took 7 days.
