Chapter 2.1, Problem 40E

### Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203

Chapter
Section

### Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203
Textbook Problem

# T-Shirt Profit The latest demand equation for your Yoda vs. Alien T-shirts is given by q = − 40 x + 600 , where q is the number of shirts you can sell in one week if you charge $x per shirt. The Student Council charges you$400 per week for use of their facilities, and the T-shirts cost you $5 each. Find the weekly cost as a function of the unit price x. Hence, find the weekly profit as a function of x, and determine the unit price you should charge to obtain the largest possible weekly profit. What is the largest possible weekly profit? To determine To calculate: The weekly cost as a function of the unit price x if the given demand equation for Yoda vs. Alien T-shirts is q=40x+600 where q is the number of T-shirts that sold per week and x is the charge per T-shirt. And, find the weekly profit as a function of x and determine the unit price that would lead to largest weekly profit. Also, the largest weekly profit. Explanation Given Information: The provided linear demand equation is, q=ā40x+600 Where, q is the number of T-shirts that can be sell in one week and x is the charges per T-shirt. The student council charges$400 per week for the use of facilities and cost of one T-shirt is $5. Formula Used: The relationship between Revenue and price is, Revenue=PriceĆQuantity The relationship between Revenue and profit is, Profit=RevenueāTotalāCost The standard parabolic function is, f(x)=ax2+bx+c Where, a, b and c are arbitrary constants. If a<0, then the corresponding parabola will be downward facing and vertex will be the highest point and if a>0, then the corresponding parabola will be upward facing and vertex will be the lowest point. And, the formula for x- coordinate of vertex is, x=āb2a Calculation: Consider the linear demand equation, q=ā40x+600 As the Student council charges$400 per week and the cost of one T-shirt is \$5.

Therefore, the total cost is given by,

TotalĀ Cost=CostĀ ofĀ facilities+(CostĀ ofĀ oneĀ t-shirt)Ć(No.Ā ofĀ T-shirtsĀ soldĀ perĀ week)C(x)=400+(5)Ć(q)

Now, substitute q=ā40x+600 in the above equation,

C(x)=400+(5)(ā40x+600)=400ā200x+3000=ā200x+3400

Therefore, total weekly cost as a function of unit price x is, C=ā200x+3400.

Now, the relationship between Revenue and Price is,

Revenue=PriceĆQuantityR(x)=(x)(q)

Substitute ā40x+600 for q in the revenue equation,

R(x)=x(ā40x+600)=ā40x2+600x

Therefore, the total Revenue R as a function of the unit price x for the given demand equation is R(x)=ā40x2+600x

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