The weekly demand for the Pulsar 25 color LED television is represented by p , w

Question

The weekly demand for the Pulsar 25 color LED television is represented by p, where p denotes the wholesale unit price in dollars and x denotes the quantity demanded.

The weekly total cost function associated with manufacturing the Pulsar 25 is given by C(x), where C(x) denotes the total cost incurred in producing x sets.

(a) Find the revenue function R.
R(x) =

Find the profit function P.
P(x) =

(b) Find the marginal cost function C '.
C '(x) =

Find the marginal revenue function R '.
R '(x) =

Find the marginal profit function P '.
P '(x) =

(c) Compute the following values. (Round your answers to two decimal places.)

Explanation / Answer

p = 610 - 0.05x , 0 <= x <= 12000

C(x) = 0.000002x3 - 0.03x2 + 510x + 74000

a) Revenue R(x) = p*x ==> R(x) = 610x - 0.05x2

Profit P(x) = Revenue R(x) - Cost C(x)

==> P(x) = 610x - 0.05x2 - (0.000002x3 - 0.03x2 + 510x + 74000)

==> P(x) = 610x - 0.05x2 - 0.000002x3 + 0.03x2 - 510x - 74000

==> P(x) = 100x - 0.02x2 - 0.000002x3 - 74000

b) Marginal cost C '(x) = 0.000002 (3) x3-1 - 0.03 (2)x2-1 + 510(1) + 0   since d/dx xn = n xn-1

==> C '(x) = 0.000006x2 - 0.06x + 510)

Marginal Revenue R '(x) = 610(1) - 0.05(2)x2-1

==> R '(x) = 610 - 0.1x

Marginal Profit P '(x) = 100(1) - 0.02(2)x2-1 - 0.000002(3)x3-1 - 0

==> P '(x) = 100 - 0.04x - 0.000006x2

c) C '(2700) = 0.000006(2700)2 - 0.06(2700) + 510 = 391.74

R '(2700) = 610 - 0.1(2700) = 340

P '(2700) = 100 - 0.04(2700) - 0.000006(2700)2 = -51.74

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