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.

Z (p) = - 10p² + 600p - 6500

Step-by-step 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.