Studdy AI Logo
Math

QuestionFind the weekly profit function P(x)P(x) for T-shirts with demand q=−50x+7200q=-50x+7200 and cost C(x)=−1800x+340200C(x)=-1800x+340200. Also, find the break-even price.

Studdy Solution

STEP 1

Assumptions1. The demand equation is given by q=−50x+7200q=-50x+7200 where qq is the number of shirts sold in a week and xx is the price per shirt. . The cost function is given by C(x)=−1800x+340200C(x)=-1800x+340200 where C(x)C(x) is the weekly cost and xx is the price per shirt.
3. The profit function is given by (x)=R(x)−C(x)(x)=R(x)-C(x) where R(x)R(x) is the revenue, C(x)C(x) is the cost and xx is the price per shirt.
4. We are not given a revenue function, but we know that revenue is the product of the price per shirt and the number of shirts sold, i.e., R(x)=xqR(x)=xq.

STEP 2

First, we need to find the revenue function R(x)R(x). We can do this by multiplying the price per shirt xx with the demand equation qq.
R(x)=xqR(x) = xq

STEP 3

Now, plug in the given demand equation for qq to calculate the revenue function.
R(x)=x(−50x+7200)R(x) = x(-50x+7200)

STEP 4

implify the revenue function.
R(x)=−50x2+7200xR(x) = -50x^2+7200x

STEP 5

Now that we have the revenue function, we can find the profit function (x)(x) by subtracting the cost function C(x)C(x) from the revenue function R(x)R(x).
(x)=R(x)−C(x)(x) = R(x) - C(x)

STEP 6

Plug in the values for the revenue function and the cost function to calculate the profit function.
(x)=(−50x2+7200x)−(−1800x+340200)(x) = (-50x^2+7200x) - (-1800x+340200)

STEP 7

implify the profit function.
(x)=−50x2+7200x+1800x−340200(x) = -50x^2+7200x+1800x-340200

STEP 8

Combine like terms in the profit function.
(x)=−50x2+9000x−340200(x) = -50x^2+9000x-340200This is the weekly profit as a function of the price per shirt xx.

STEP 9

To break even, the profit must be zero. So, we set the profit function equal to zero and solve for xx.
−50x2+900x−340200=-50x^2+900x-340200 =

STEP 10

This is a quadratic equation in the form ax2+bx+c=0ax^2+bx+c=0. We can solve it using the quadratic formula x=−b±b2−4ac2ax=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.

STEP 11

Plug in the values for aa, bb, and cc into the quadratic formula to solve for xx.
x=−9000±9000−4(−50)(−340200)(−50)x=\frac{-9000\pm\sqrt{9000^-4(-50)(-340200)}}{(-50)}

STEP 12

implify the expression under the square root.
x=−9000±81000000−68040000−100x=\frac{-9000\pm\sqrt{81000000-68040000}}{-100}

STEP 13

Calculate the value under the square root.
x=−9000±12960000−100x=\frac{-9000\pm\sqrt{12960000}}{-100}

STEP 14

Take the square root.
x=−9000±3600−100x=\frac{-9000\pm3600}{-100}

STEP 15

olve for xx.
x=−9000+3600−100 or x=−9000−3600−100x=\frac{-9000+3600}{-100} \text{ or } x=\frac{-9000-3600}{-100}

STEP 16

Calculate the values for xx.
x=54 or x=126x=54 \text{ or } x=126So, you break even when you charge either 54or54 or 126 per shirt. When you set the price per shirt to one of these values, your profit is $0.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord