The demand equation for your company\'s virtual reality video headsets is p = 1,

Question

The demand equation for your company's virtual reality video headsets is p = 1,500/q0.3where q is the total number of headsets that your company can sell in a week at a price of p dollars. The total manufacturing and shipping cost amounts to $80 per headset.

(a) Find the weekly cost, revenue and profit as a function of the demand q for headsets.

C(q)=

R(q)=

P(q)=

(b) How many headsets should your company sell to maximize profit? (Give your answer to the nearest whole number.)

q=_____ headsets

What is the greatest profit your company can make in a week? (Give your answer to the nearest whole number.)

$_____

Second derivative test: Your answer above is a critical point for the weekly profit function. To show it is a maximum, calculate the second derivative of the profit function.

P"(q)=

Explanation / Answer

We have:

p=1500/q0.3

Part (a)

Cost C (q) = 80q

Revenue R (q) = pq = (1500/q0.3)q = 1500q0.7

Profit P (q) = R-C = 1500q0.7-80q

Part (b)

For maximum profit we set the first derivative of profit equal to zero:

P'(q) = 1500 (0.7q-0.3)-80 = 0

1050q-0.3=80

q0.3=1050/80 = 105/8

q = (105/8)(10/3)

q = 5333.32

Therefore, on rounding we get q=5333 headsets

Part (c)

Greatest profit = 1500 (5333)0.7-80*5533

Greatest profit = 182857 dollars

P''(q)=1050 (-0.3)q-1.3

P''(5333) = -315 (5333)-1.3

P"(5333)=-0.000014

Therefore, negative value double derivative implies profit is maximum.

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