Question

A marine biologist is studying the growth of a particular species of fish. She writes the following equation to show the length of the fish, f(m), in cm, after m months:f(m) = 4(1.08)mPart A: When the marine biologist concluded her study, the length of the fish was approximately 6.86 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(m) represent? Part C: What is the average rate of change of the function f(m) from m = 3 to m = 7, and what does it represent?

Answer 1

Explanation:

Where m is the number of months and f(m) is the length of the fish in cm.

a. When the study ended the function had a value f(m)=6.86 cm.

To find the domain for the growth function, we need to equal the equation to 6.86 and solve for m, as follows:

Thus, a reasonable domain to plot the growth function is (0,7).

b. The y-intercept of the graph of the function f(m) represents the length of the fish when the study started.

c. The average rate of change of an exponential function is given by the following formula:

Where (x1, y1) and (x2,y2) are the coordinates of the points at m=3 and m=7.

First, let's solve the function at m=3:

When m=7:

Thus the coordinates are (3,5.04) and (7,6.86), by replacing these values in the formula we obtain:

The average rate of change is 0.45 in the interval (3,7). It represents the growth rate in cm per month

Answer 2

The given equation is:

`f(m)=4(1.08)^m`

Where m is the number of months and f(m) is the length of the fish in cm.

a. When the study ended the function had a value f(m)=6.86 cm.

To find the domain for the growth function, we need to equal the equation to 6.86 and solve for m, as follows:

`\begin{gathered} 4(1.08)^m=6.86 \\ 1.08^m=(6.86)/(4) \\ 1.08^m=1.715 \\ \text{Apply log on both sides} \\ \log (1.08)^m=\log (1.715) \\ \text{Apply the power of logs} \\ m\cdot\log (1.08)=\log (1.715) \\ m=(\log 1.715)/(\log 1.08) \\ m=(0.234)/(0.033) \\ m=7months \end{gathered}`

Thus, a reasonable domain to plot the growth function is (0,7).

b. The y-intercept of the graph of the function f(m) represents the length of the fish when the study started.

c. The average rate of change of an exponential function is given by the following formula:

`m=(y_2-y_1)/(x_2-x_1)`

Where (x1, y1) and (x2,y2) are the coordinates of the points at m=3 and m=7.

First, let's solve the function at m=3:

`\begin{gathered} f(3)=4(1.08)^3 \\ f(3)=4\cdot1.26 \\ f(3)=5.04 \end{gathered}`

When m=7:

`\begin{gathered} f(7)=4(1.08)^7 \\ f(7)=4\cdot1.71 \\ f(7)=6.86 \end{gathered}`

Thus the coordinates are (3,5.04) and (7,6.86), by replacing these values in the formula we obtain:

`m=(6.86-5.04)/(7-3)=(1.82)/(4)=0.45`

The average rate of change is 0.45 in the interval (3,7). It represents the growth rate in cm per month.