Question

A manufacturer can produce digital recorders at a cost of 50 dollars each. It is estimated that if the recorders are sold for p dollars apiece, consumers will buy q=120p recorders each month. a) Express the manufacturer's profit P as a function of q.
b)What is the average rate of profit obtained as the level of production increases from q=0 to q=15? (at dollars per unit)
c) At what rate is profit changing when q=15 recorders are produced? (at dollars per unit)

Answer 1

Answer:

a. P(q) = 70q - q²

b. Average rate of Profit = 55

c. Rate of profit changing = 40 units per dollar

Step-by-step explanation:

a.

Recall that Profit = Revenue - Cost

Cost Preorder = $50

So, cost = 50q.

Since p is the revenue price for 1 unit,

We get that the revenue which is retail price per unit.

So, revenue = pq

Recall that q = 120-p ------ make p the subject of formula

p = 120-q

So, Revenue = (120-q) * q =

Profit = Revenue - cost

The Equation of profit P, in terms of q is given by

P(q) = (120 - q)q - 50q

P(q) = 120q - q² - 50q.

P(q) = 70q - q²

b.

At q = 0, P(q) = 0

At q = 15

P(q) = (120-15)15 - 50*15

P(q) = 105*15-50*15

P(q) = 825

Average Rate of change = ∆Profit/∆Quantity

Average Rate of change = (825-0)/(15-0)

Average Rate of change = 55

c.

To calculate the rate of Profit change, we need to use the limit definition to find the derivative of

P(q) = 70 - q²

Limit definition of the derivative = Lim h->0 ( f(x+h) - f(x) ) / h

Substitute the value la of p(q) and p(q+h) in the above

P'(q)Lim h->0 = (P(q+h) - P(q) )/ h

P'(q)Lim h->0 =( [70(q+h) - (q+h)²] - [70q - q²] ) / h

P'(q)Lim h->0 = (70q + 70h - q² - 2qh - h² - 70q + q²)/h

P'(q)Lim h->0 = (70h - 2qh - h²)/h

P'(q)Lim h->0 = 70 - 2q - h

= 70 - 2q - (0)

= 70 - 2q

When q = 15

70 - 2(q) becomes

= 70 - 2(15)

= 70 - 30

= 40 units per dollar