Question

NetSell, a the TV remote control supplier for Lumyn Electronics, has a weekly production cost of q TV remote controls that is given by

C(q) = 0.000004q^3 - 0.03q^2 + 100q + 75,000

where q is in the interval [0, 10,000].

The demand function for this product is given by

p(q) = -0.005q + 200.

Based on this information, find the following:

a) The marginal cost for the company.

b) The marginal revenue for the company.

c) The marginal profit for the company when 2,000 and 7,000 TV remote controls are manufactured.

Answer 1

a.
`(dC(q))/(dq) = 0.000012q^2 -0.06q + 100`

b.
`(dR(q))/(dq)=-0.01q+200`

c.

`U'(2000)=-0.000012(2000)^2+0.05(2000)+100 = 152`

`U'(7000)=-0.000012(7000)^2+0.05(7000)+100 = -138`

Explanation:

a) The marginal cost function is given by the derivative of the total cost function, in this way the marginal cost function for this company is:

`(dC(q))/(dq) = (dC(q))/(dq) (0.000004q^ 3 - 0.03q ^ 2 + 100q + 75000) = 0.000012q^2 -0.06q + 100`

b) The income function is given by the relation
`R (q) = P (q) q = -0.005q^2 + 200q`.

The marginal revenue function for the company is given by the derivative of the revenue function, in this way the marginal revenue function is:

`(dR(q))/(dq)= -0.01q+200`

(c) The profit function of the company is given by the relation
`U (q) = R (q) - C (q)`, and the marginal utility function is given by the derivative of the utility function, in this way , the marginal utility function is:

`(dU(q))/(dq)=R'(q) - C'(q) = -0.01q+200 - (0.000012q^2-0.06q+100) = -0.000012q^2+0.05q+100`

When q = 2000, the marginal utility is:

`U'(2000)=-0.000012(2000)^2+0.05(2000)+100 = 152`

When q = 7000, the marginal utility is:

`U'(7000)=-0.000012(7000)^2+0.05(7000)+100 = -138`