Chapter 4.8, Problem 28E

(a)

To determine

How many weeks should Mr. Jackson wait in order to maximize the profit?

Answer to Problem 28E

  $2\text{ weeks}$

Explanation of Solution

Given information: Agriculture: If Wesley Jackson harvests his apple crop now; the yield will average 120 pounds per tree. Also, he will be able to sell the apples for $0.48 per pound. However, he knows that if he waits, his yield will increase by about 10 pounds per week, while the selling price will decrease by $0.03 per pound per week.

  

Calculation:

If $\begin{array}{l}x=\text{weeks}\\ P=\left(0.48-0.03x\right)\left(120+10x\right)\\ P=-0.3x^2+1.2x+57.6\end{array}$

In this form the vertex is at:

  $\left(-\frac{b}{2a},f\left(-\frac{b}{2a}\right)\right)$

  $\begin{array}{l}-\frac{b}{2a}=-\frac{1.2}{2\left(-0.3\right)}\\ =2\text{ weeks}\end{array}$

(b)

To determine

What is the maximum profit?

Answer to Problem 28E

They should charge 58.80 per tree

Explanation of Solution

Given information: Agriculture: If Wesley Jackson harvests his apple crop now; the yield will average 120 pounds per tree. Also, he will be able to sell the apples for $0.48 per pound. However, he knows that if he waits, his yield will increase by about 10 pounds per week, while the selling price will decrease by $0.03 per pound per week.

  

Calculation:

  $P\left(2\right)=-0.3{\left(2\right)}^2+1.2\left(2\right)+57.6=58.80$

They should charge 58.80 per tree.