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? (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 = 1 to n = 5, and what does it represent? (4 points)

Answer 1

Answer:

(A) The reasonable domain to plot the growth function is (0 ≤ n < 5).

(B) The y-intercept of the function is 10 cm. It represents the height of the plant at the beginning of the experiment.

(C) The slope of 0.21 implies that the plant is growing 0.21 cm every day.

Step-by-step explanation:

The equation representing the height of the plant is:

$f (n)=10\cdot (1.02)^{n}$

(A)

It is provided that at the end of the study the height of the plant was approximately 11.04 cm.

Compute the range of values of n as follows:

$f (n)=10\cdot (1.02)^{n}\\11.04 = 10\cdot(1.02)^{n}\\1.104 = 1.02^{n}\\\ln (1.104) = \ln (1.02^{n})\\\ln (1.104) = n \ln (1.02)\\\\n = \frac{\ln (1.104)}{ \ln (1.02)}\\\\n = 4.99630409516\\n\approx 5$

Thus, the reasonable domain to plot the growth function is (0 ≤ n < 5).

(B)

The y-intercept, in this case would represents the height of the plant when they began the experiment.

That is, the y-intercept of the function is the value of f (n) for n = 0.

Compute the value of f (n = 0) as follows:

$f (0)=10\cdot (1.02)^{0}=10$

Thus, the y-intercept of the function is 10 cm. It represents the height of the plant at the beginning of the experiment.

(C)

Compute the values of f (n = 1) and f (n = 5) as follows:

$f (1)=10\cdot (1.02)^{1}=10.2\\\\f (5)=10\cdot (1.02)^{5}=11.0408\approx 11.04$

Compute the average rate of change as follows:

$\text{Average rate of change}=\frac{f(n_{2})-f(n_{1})}{n_{2}-n_{1}}$

                                    $=\frac{11.04-10.20}{5-1}\\\\=\frac{0.84}{4}\\\\=0.21$

Thus, the average rate of change of the function f(n) from n = 1 to n = 5 is 0.21.

The average rate of change is known as the slope of the function. The slope represents how much the plant is growing every day.

Thus, the slope of 0.21 implies that the plant is growing 0.21 cm every day.