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?
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
Check my thinking. Check my work.
To answer these questions, we need to build an equation to represent Mr. Jackson's profit over time. Let's break it down step by step:
Initial parameters:
- Yield at time t=0: 120 pounds per tree
- Selling price at time t=0: $0.48 per pound
- Increase in yield per week: 10 pounds
- Decrease in selling price per week: $0.03 per pound
We can represent the number of weeks Mr. Jackson waits as 'w'. Now, let's define the following variables:
- Yield at time t=w: 120 + 10w (since the yield increases by 10 pounds each week)
- Selling price at time t=w: 0.48 - 0.03w (since the selling price decreases by $0.03 per pound each week)
To calculate the profit, we multiply the yield by the selling price:
Profit at time t=w: (120 + 10w) * (0.48 - 0.03w)
a. To determine the number of weeks that will maximize profit, we need to find the maximum point of the profit equation. We can do this by finding the derivative of the profit equation, setting it equal to zero, and solving for 'w'.
Let's calculate the derivative and set it to zero:
Profit derivative: d/dw [(120 + 10w) * (0.48 - 0.03w)]
Setting the derivative equal to zero: (120 + 10w) * (-0.03) + (0.48 - 0.03w) * 10 = 0
Now, solve this equation for 'w' to find the number of weeks that will maximize profit.
b. Once you have the value of 'w' from part (a), substitute it into the profit equation to find the maximum profit.
This approach will enable us to determine the number of weeks Mr. Jackson should wait and the corresponding maximum profit.