Question

Given that weekly demand curve of a local wine producer is p = 50 − 0.1q, and that

the total cost function is C(q) = 1500 + 10q, where q bottles are produced each
day and sold at a price of $p per unit.
a. Express the weekly profit as a function of price p.

Answer 1

Z(p) = -10p² + 600p - 6500

Explanation:

Demand curve p = 50 - 0.1 q

Total cost function C(q) = 1500 + 10q

where q is the number of bottles produced each day and p is the selling prices per bottle.

Now, p = 50 - 0.1 q

o.1 q = 50 - p

q = 500 - 10p

Revenue = price × quantity = p × q

= p ( 500 - 10p)

= 500 p - 10p²

Profit = total revenue - total cost

Let Profit be Z since profit is function of price, therefore

Z(p) = pq - C(q)

Z (p)= 500 p - 10p² - (1500 + 10q)

Substituting the value of q in above expression,

Z(p) = 500 p - 10p² - ( 1500 + 10 (500 - 10p))

Z(p) = 500p -10p² - 1500 -5000 + 100p

Z(p) = -10p² + 600p - 6500

So, the weekly profit as a function of price p is Z(p) = -10p² + 600p - 6500.