A farmer is tracking the amount of corn his land is yielding each year. He finds that the function f(x) = −x2 + 20x + 50 models the crops in pounds per acre over x years. Find and interpret the average rate of change from year 1 to year 10.

Question

A farmer is tracking the amount of corn his land is yielding each year. He finds that the function f(x) = −x2 + 20x + 50 models the crops in pounds per acre over x years. Find and interpret the average rate of change from year 1 to year 10.

Answer 1

To find the average rate of change from year 1 to year 10, we need to find the change in the function value over this time period.

First, we find the function values for year 1 and year 10:

For year 1: f(1) = -1^2 + 20(1) + 50 = 69 pounds per acre

For year 10: f(10) = -10^2 + 20(10) + 50 = 150 pounds per acre

Now we can find the change in function value over this time period:

Change in function value = f(10) - f(1) = 150 - 69 = 81 pounds per acre

Finally, we can find the average rate of change by dividing the change in function value by the change in years:

Average rate of change = Change in function value / Change in years = 81 pounds per acre / 9 years = 9 pounds per acre per year

Interpretation: The average rate of change from year 1 to year 10 is 9 pounds per acre per year. This means that, on average, the amount of corn produced per acre of land increased by 9 pounds each year over this time period.