Question

question 6(multiple choice worth 1 points) (08.01 mc) a farmer is tracking the number of soybeans his land is yielding each year. he finds that the function f(x)=-x² + 20x + 100 models the crops in pounds per acre over x years. find and interpret the average rate of change from year 10 to year 20. the crop yield increased by 200 pounds per acre from year 10 to year 20. the crop yield increased by 100 pounds per acre from year 10 to year 20. the crop yield decreased by 10 pounds per acre from year 10 to year 20. the crop yield decreased by 0.1 pounds per acre from year 10 to year 20.

Explanation:

Step1: Recall average rate of change formula

The average rate of change of a function $y = f(x)$ from $x = a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a = 10$, $b=20$ and $f(x)=-x^{2}+20x + 100$.

Step2: Calculate $f(10)$

Substitute $x = 10$ into $f(x)$:
$f(10)=-(10)^{2}+20\times10 + 100=-100 + 200+100=200$.

Step3: Calculate $f(20)$

Substitute $x = 20$ into $f(x)$:
$f(20)=-(20)^{2}+20\times20 + 100=-400+400 + 100=100$.

Step4: Calculate average rate of change

$\frac{f(20)-f(10)}{20 - 10}=\frac{100 - 200}{10}=\frac{-100}{10}=- 10$.

Answer:

The crop yield decreased by 10 pounds per acre from year 10 to year 20.