The weekly demand for the Pulsar 40-in. high-definition television is given by t

Question

The weekly demand for the Pulsar 40-in. high-definition television is given by the demand equation

where p denotes the wholesale unit price in dollars and x denotes the quantity demanded. The weekly total cost function associated with manufacturing these sets is given by

where C(x) denotes the total cost incurred in producing x sets. Find the level of production that will yield a maximum profit for the manufacturer. Hint: Use the quadratic formula. (Round your answer to the nearest whole number.)

__ Units

The weekly demand for the Pulsar 40-in. high-definition television is given by the demand equation where C(x) denotes the total cost incurred in producing x sets. Find the level of production that will yield a maximum profit for the manufacturer. Hint: Use the quadratic formula. (Round your answer to the nearest whole number.) __ Units where p denotes the wholesale unit price in dollars and x denotes the quantity demanded. The weekly total cost function associated with manufacturing these sets is given by

Explanation / Answer

Demand equation : p = -0.06x + 649

The revenue gained = Demand * p

R(x) = x(-0.06x + 649)

R(x) = -0.06x^2 + 649x

C(x) = 0.000003x^3 - 0.02x^2 + 400x + 80000

Now, profit is :

R - C, which becomes :

-0.06x^2 + 649x - ( 0.000003x^3 - 0.02x^2 + 400x + 80000)

P(x) = -0.000003x^3 - 0.04x^2 + 249x - 80000

Now, lets derive this to find the critical numbers

P'(x) = -0.000009x^2 - 0.08x + 249 = 0

Solving that by quadratic formula, we get :

x = -11330.6 or 2441.76

Obviously x cannot be negative

So, x = 2441.76

Lets confirm that this results in a MAXIMUM

To do this, we find the second derivative

P''(x) = -0.000018x - 0.08

P''(2441.76) = -0.000018(2441.76) - 0.08 --> some negative number

Since the second derivative is negative, the critical number x = 2441.76 corresponds to a MAXIMAL case

x =2441.76

Rounding, w eget x = 2442

So, 2442 units -----> ANSWER

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