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
