Question

The latest demand equation for your Banjos Rock T-shirts is given by q = −30x + 7200 where q is the number of shirts you can sell in one week if you charge x dollars per shirt. When you charge x dollars per shirt, your weekly cost function (in dollars) is given by C(x) = −1800x + 567000 (a) Find the weekly profit as a function of the price per shirt x.

Answer 1

Answer:

P(x)=-30x^2+9000x-567000

Explanation:

First, we need to remember the parts of a Profit function. A profit a business makes  equals revenue (R(x)) minus its costs (C(x)). So

$P(x)=R(x)-C(x)$

There are two parts

1. Revenue: which is equal to the number of units sold times the price:

$R(x)=Q(x)\times x$

where x is the price you charge and Q(x) is the number of shirts that can be sold. Then

$R(x)=(-30x+7200)\times x=7200x-30x^2$

2. Cost. The cost function is directly given by the question

$C(x)=567000-1800x$

Putting this together we have

$P(x)=R(x)-C(x)=7200x-30x^2-567000+1800x\\P(x)=-30x^2+9000x-567000$

Answer 2

The profit as a function of the price per shirt is:

$\rm -30x^2 + 9000x - 56,700$.

How to calculate profit function?

Given:

$\rm Demand\: Equation Q(x) = -30x +7,200\\\\\rm Cost\: function \:C(x) = -1,800x + 56,700$

The profit can be calculated as a subtraction of cost from total revenue.

Total revenue is the product of quantity demand and price.

$\rm Total\ revenue= Q(x)\times x\\\\\rm Total\ revenue = (-30x +7,200)x\\\\\rm Total\ revenue= -30x^2 + 7,200x$

Profit is the difference between revenue and cost. Therefore,

$\rm Profit = R(x) - C(x)\\\\P(x) = (-30x^2 +7,200x) - (-1800x + 56,700)\\\\P(x) = -30x^2 +7,200x + 1800x - 56,700\\\\P(x) = -30x^2 + 9,000x - 56,700$

Hence the profit function is:

$\rm -30x^2 + 9000x - 56,700$

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