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. The crop yield increased by 9 pounds per acre from year 1 to year 10. The crop yield decreased by 0.09 pounds per acre from year 1 to year 10. The crop yield decreased by 11 pounds per acre from year 1 to year 10. The crop yield increased by 99 pounds per acre from year 1 to year 10.

Answer 1

Answer: The crop yield increased by 9 pounds per acre from year 1 to year 10.

Step-by-step explanation:

To solve this we are using the average rate of change formula: Av=\frac{f(x_2)-f(x_1)}{x_2-x_1}, where:

x_2 is the second point in the function

x_1 is the first point in the function

f(x_2) is the function evaluated at the second point

f(x_1) is the function evaluated at the first point

We know that the first point is 1 year and the second point is 10 years, so x_1=1 and x_2=10. Replacing values:

Av=\frac{-(10)^2+20(10)+50-[-(1)^2+20(1)+50]}{10-1}

Av=\frac{-100+200+50-[-1+20+50]}{9}

Av=\frac{150-[69]}{9}

Av=\frac{150-69}{9}

Av=\frac{81}{9}

Av=9

Since f(x) represents the number of pounds per acre and x the number of years, we can conclude that the crop yield increased by 9 pounds per acre from year 1 to year 10.

Answer 2

Answer:

The crop yield decreased by 0.09 pounds per acre from year 1 to year 10

Step-by-step explanation:

The function that models the crops in pounds per acre over x years is

$f(x)=-x^2+20x+50$

In year 1, the crop yield is $f(1)=-(1)^2+20(1)+50=69$

In year 10, the crop yield is $f(10)=-(10)^2+20(10)+50=150$

The average rate of change from year 1 to year 10 is given by:

$\frac{f(10)-f(1)}{10-1}$

We substitute the values to get:

$\frac{150-69}{10-1}$

$=\frac{81}{9}=9$

Since the average rate of change is positive, the crop yield decreased by 0.09 pounds per acre from year 1 to year 10.