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Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203

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BuyFindarrow_forward

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=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=Price×Quantity

The relationship between Revenue and profit is,

Profit=RevenueTotalCost

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=b2a

Calculation:

Consider the linear demand equation,

q=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,

Total Cost=Cost of facilities+(Cost of one t-shirt)×(No. of T-shirts sold per week)C(x)=400+(5)×(q)

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

C(x)=400+(5)(40x+600)=400200x+3000=200x+3400

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

Now, the relationship between Revenue and Price is,

Revenue=Price×QuantityR(x)=(x)(q)

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

R(x)=x(40x+600)=40x2+600x

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

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