For the cost and price functions below, find



a. the number, q, of units that produces maximum profit



b. the price, p, per unit that produces maximum profit



c. the maximum profit, P.



C(q) = 70 + 17q



p = 77 - 2q

Answer:

a) The number, q, of units that produces maximum profit = 15

b) The price, p, per unit that produces maximum profit = 47 (currency not giben in the question)

c) Maximum Profit = P = 380 (currency not given in the question).

Explanation:

The cost function and price per unit function are given respectively as

C(q) = 70 + 17q

p = 77 - 2q

where q = quantity or number of units

a.) the number, q, of units that produces maximum profit

Total cost = C(q) = 70 + 17q

Revenue = (price per unit) × (Number of units) = p × q = (77 - 2q) × q = (77q - 2q²)

Profits = P(q) = (Revenue) - (Total Cost)

P(q) = (77q - 2q²) - (70 + 17q)

P(q) = -2q² + 60q - 70

To maximize the profits, we just obtain the point where the profit function reaches a Maximum.

At the maximum of a function, (dP/dq) = 0 and (d²P/dq²) < 0

Profit = P(q) = -2q² + 60q - 70

(dP/dq) = -4q + 60

At maximum point,

(dP/dq) = -4q + 60 = 0

q = (60/4) = 15

(d²P/dQ²) = -4 < 0 (hence, showing that the this point corresponds to a maximum point truly)

Hence, the number, q, of units that produces maximum profit = 15.

b.) the price, p, per unit that produces maximum profit

The price per unit is given as

p = 77 - 2q

Maximum profit occurs at q = 15

p = 77 - (2×15) = 47

Hence, the price, p, per unit that produces maximum profit = 47 (currency not given in the question)

c.) the maximum profit, P.

The Profit function is given as

Profit = P(q) = -2q² + 60q - 70

At maximum Profit, q = 15

Maximum Profit = P(15)

= -2(15²) + 60(15) - 70

= 380 (currency not given in the question).

Hope this Helps!!!

A) The number, q, of units that produce maximum profit is = 15

B) The price, p, per unit that creates maximum profit is = 47

C) Maximum Profit is = P = 380

When The cost procedure and price per unit procedure are presented respectively as:

C(q) is = 70 + 17q

p is = 77 - 2q

where that q is = quantity or number of units

a.) When the number, q, of units that produce maximum profit

The Total cost is = C(q) = 70 + 17q

When the Revenue is = (price per unit) × (Number of units) that is = p × q = (77 - 2q) × q is = (77q - 2q²)

After that Profits is = P(q) = (Revenue) - (Total Cost)

Then P(q) is = (77q - 2q²) - (70 + 17q)

Now, P(q) is = -2q² + 60q - 70

When To maximize the profits, Then we just obtain the point where the profit function reaches a Maximum.

When At the maximum of a function, (dP/dq) is = 0 and (d²P/dq²) < 0

Profit is = P(q) = -2q² + 60q - 70

(dP/dq) is = -4q + 60

Then At maximum point are:

(dP/dq) is = -4q + 60 = 0

After that, q = (60/4) = 15

Then (d²P/dQ²) = -4 < 0 (hence, showing that this point corresponds to a maximum point truly)

Therefore, the number, q, of units that produce maximum profit is = 15.

b.) When the price, p, per unit that produces maximum profit

The price per unit is given as

p is = 77 - 2q

Then Maximum profit occurs at q is = 15

p is = 77 - (2×15) = 47

Therefore, the price, p, per unit that produces maximum profit is = 47 (currency not provided in the question)

c.) When the maximum profit, P.

The Profit function is given as

Profit is = P(q) = -2q² + 60q - 70

Then At maximum Profit, q = 15

So, The Maximum Profit is = P(15)

Then = -2(15²) + 60(15) - 70

Therefore, = 380 (currency not given in the question).

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