Answered

IDNLearn.com: Where your questions meet expert advice and community support. Find the information you need quickly and easily with our reliable and thorough Q&A platform.

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

Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! Thank you for choosing IDNLearn.com for your queries. We’re here to provide accurate answers, so visit us again soon.