Optimization in economics help

[awmonster]

Homework Statement

A grocer usually buys 30 dozen pineapples per week. The cost per pinapple is $0.40 and they are sold at 0.78 a piece. As there is an abundant crop on May, the wholesaler offers a discount of $0.05 per pineapple for each additional order of 5 dozens (i.e. if the grocer buys 35 dozen, then the cost per pineapple is $0.35) In order to ensure that all of these perishable items are sold, the grocer decides to decrease the unit price by $0.08 for each additional 5 dozen that are bought (i.e. if the grocer buys 35 dozen, then the price per pineapple is $0.70) How many dozen pineapples should the grocer buy in order to maximize the profit?

Homework Equations

Profit=(sale price-cost price)(x number of pineapples sold)
y= number of 5 dozen additional pinapples, i.e. if grocer buys 35 dozen y=1, 40 dozen then y=2

The Attempt at a Solution

Sale Price=0.78-0.08y
Cost Price=0.40-0.05y
Profit=x(0.38-0.03y)
I am lost after this...

 

[mighty2000]

I would suggest approaching this problem using a mathematical model to determine the optimal number of dozen pineapples to buy in order to maximize profit.

First, let's define some variables:
- x: number of dozen pineapples bought (we will assume this is a whole number)
- y: number of additional orders of 5 dozen pineapples (we will assume this is a whole number as well)
- C: cost per pineapple
- P: price per pineapple
- R: revenue (total amount received from selling pineapples)
- D: total number of pineapples bought (D = 12x)

Now, we can write the following equations:
C = 0.40 - 0.05y
P = 0.78 - 0.08y
R = P*D = (0.78 - 0.08y)*12x = 9.36x - 0.96xy

To maximize profit, we need to find the value of x that will give us the maximum value of R. To do this, we can take the derivative of R with respect to x and set it equal to 0:
dR/dx = 9.36 - 0.96y = 0
9.36 = 0.96y
y = 9.75

Now, we can plug this value of y back into our equation for P to find the optimal price per pineapple:
P = 0.78 - 0.08*9.75 = 0.08

This means that the grocer should buy 9 dozen pineapples (108 pineapples) and offer a price of $0.08 per pineapple to maximize profit.

I hope this helps you in your problem-solving process. Best of luck!