Question

Seth has a different team of volunteers search another region for the plant species. He has already found 30 plants himself, and the number of plants found increases by 16% for each new volunteer added to the search.

Part A
Find a function that relates the number p of plants found to the number x of volunteers in the search.

Part B
What is the average rate of change from 0 to 5 volunteers? Round to the nearest whole number.

Answer 1

Explanation:

Part A:

Let's start with the initial number of plants found by Seth, which is 30. For each new volunteer added to the search, the number of plants found increases by 16%.

So, if x represents the number of volunteers, then the number of plants found can be calculated using the following function:

p(x) = 30(1.16)^x

This function gives us the number of plants found as a function of the number of volunteers in the search.

Part B:

To find the average rate of change from 0 to 5 volunteers, we need to calculate the slope of the function p(x) over that interval.

The slope of a function is given by the formula:

slope = (change in y) / (change in x)

In this case, the change in x is 5 - 0 = 5 (since we're looking at the interval from 0 to 5 volunteers).

To find the change in y, we can plug in x=5 and x=0 into the function p(x) and subtract:

p(5) - p(0) = 30(1.16)^5 - 30(1.16)^0

p(5) - p(0) = 70.43 - 30

p(5) - p(0) = 40.43

So, the change in y over the interval from 0 to 5 volunteers is 40.43.

Therefore, the average rate of change from 0 to 5 volunteers is:

slope = (change in y) / (change in x) = 40.43 / 5 = 8.086

Rounding to the nearest whole number, the average rate of change from 0 to 5 volunteers is 8.