Question

Wally's T-shirt company estimates the demand function for the Red sox world series t-shirt to be:

q=-4p+200

where q is the number of shirts you can sell in one week if you charge sx 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?

Answer 1

Answer:costC(p)= 1400+20p

R(p)= 4p^2+200p

Profit= 4p^2+ 180p-1400

Step-by-step explanation:

q=4p+200

Student council charge $400 per week.

Revenue= price × quantity.

Relationship between revenue and profit= Revenue-total cost.

Total cost= cost of facilities +(cost of one shirt)× number of T-shirts sold per week.

C(p)= 400 + 5(q)

Put q=4p+200

C(p)= 400 + 5 (4p +200)

C(p)= 400+ 20p + 1000

C(p)= 1400+ 20p

Relationship between revenue and price=Revenue= price × quantity

R(p)= p×q

Put q=4p +200

R(p)= p ×(4p +200)

R(p)= 4p^2+ 200p

Profit= Revenue - total cos

Profit=R(p) -C(p)

Profit= (4p^2 + 200p) - ( 20p +1400)

Profit= 4p^2 + 180p -1400

Answer 2

Answer:

(i) The cost function is C(q) =5q + 400

(ii) The profit function =$5q - q^2 /4$for weekly profit

(iii)  the maximum profit is $243.75

Step-by-step explanation:

(i) the cost function will be C(q) = FC(q) + V(q)

fixed costs FC(q) as a function of quantiyy and variable costs VC(q) we already know that the fixed cost is the $400 per week from charges to use facilities then Variable costs is $5 times the quantity of t shirts sold per week.

so C(q) = $400 + 5q

(ii) The profit function will be found by multiplying quantity demanded with the price at which that quantity is demanded at so we have a demand function q= -4p + 200 then we make the price the subject of the formula

p = 200/4 - q/4 we divided both sides by 4 after transposing -4p

then we get p= 50 - q/4 which is the price of the quantity demanded then we multiply this price by q

thereafter we get pq = 50q - q^2 /4

which is our profit function because p x q is revenue/ profit ;

$Profit = 50q - q^2 / 4$