Question

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.

a. How many weeks should Mr. Jackson wait in order to maximize the profit?
b. What is the maximum profit?

How do you write an equation to represent all of this and lead you to your answer?

Answer 1

I presume that the 10 lbs increase each week is per tree; otherwise we need the number of trees.
You can make a table for one tree.

lbs.....price/lb....week...total price
120..... 0.48 ...... now .. 57.60
130..... 0.45 ....... 1 ... 58.50
140..... 0.42 ....... 2 ... 58.80
150..... 0.39 ....... 3 ... 58.50
160..... 0.36 ....... 4 ... 57.60

Here is the way I would do the equation.
Y = # weeks
120 + 10Y = # lbs apples available.
selling price is 0.48-0.03Y
# lbs apples available x selling price is total price Mr. Jackson obtains.
(120+10Y)(0.48-0.03Y)=selling price (s.p.) each week.
57.60 + 1.20Y -0.3Y² =s.p.
You can try week 1,2,3,and 4 and see what the selling price is.
I did that and obtained
now......57.60
week 1...58.50
week 2...58.80
week 3...58.50
week 4...57.60
<b> Check my thinking. Check my work. </b>