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If you want to sell your cupcakes for $2 per dozen, you can make a profit of $160. If raise the cupcake price by
$1, you can make a profit of $216. If you raise the cupcake price by $2 (which makes them S4 per dozen) you can
make $256 profit. What is the maximum profit you can make? What should you charge for your cupcakes?

Sagot :

Facundo3141592idn

The maximum profit is $288, and you should charge $6 per dozen.

How to get the relation between demand and prize?

Let's assume that demand depends linearly on price, such that we can write:

q = a*p + b

where q is the demand and p is the price.

We have 3 points of that line:

  • When you sell at $2 per dozen, you win $160, so you sold 80 dozens.
  • When you sell at $3 per dozen, you win $216, so you sold 72 dozens.
  • When you sell at $4 per dozen, you win $256, so you sold 64 dozens.

The slope of the line is given by:

[tex]a = \frac{q_2 - q_1}{p_2 - p_1}[/tex]

Where these are values of demand and prize for two different points, if we use the first two points we get:

[tex]a = \frac{72 - 80}{3 - 2} = -8[/tex]

Then the line is something like:

q = -8*p + b

Remember that when p = 2, the demand is equal to 80, replacing that we get:

80 = -8*2 + b

80 + 16 = b = 96.

So the line is:

q = -8*p + b

How to maximize the profit?

Profit is given by the product between the demand and the price, so we have.

Profit = p*q = p*(-8*p + 96) = -8p^2 + 96p

We want to maximize that, notice that is a quadratic equation of negative leading coefficient, so the maximum is at the vertex, using the equation for the vertex of a quadratic equation we will see that the maximum is at:

p = -96/(2*-8) = 6

Then the maximum profit is:

profit = -8*(6)^2 + 96*6 = 288

The maximum profit is $288, and you should charge $6 per dozen.

If you want to learn more about linear relations, you can read:

https://brainly.com/question/1884491

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