The full shot is the problem with all questions. Since I had to redo the questio

Question

The full shot is the problem with all questions. Since I had to redo the question the new equation is at the bottom. Please answer using the new equation.

3.6.37 Question Help Let C(x) represent the cost of producing x items and p(x) be the sale price per item if x items are sold. The profit Px) of selling x items is P(x) xp(x)-Cx) (revenue minus costs). The average profit per item when x items are sold is P(x)/x and the marginal profit is dPdx. The marginal profit approximates the profit obtained by selling one more item given that x items have already been sold. Consider the following cost functions C and price functions p. Complete parts (a) through (d) below C(x)--004x2 + 40x + 100, p(x) = 100, a = 500 a. Find the profit function P The profit function is P(x) = 0.04x‘ + 60x-100 b. Find the average profit function and marginal profit function. The average profit function is-_ 0.04x + 60-- P(x) 100 dP dx The marginal profit function is :0.08x + 60 c. Find the average profit and marginal profit if x a units have been sold. The average profit if x = a units have been sold is $ 79.80.

Explanation / Answer

a)

C(x)= -0.03x² +50x +80

R(x) =x*p(x) =300x

Find profit as

P(x)= R(x) -C(x)

P(x) = 300x - (-0.03x² +50x +80)

P(x) = 0.03x² +250x -80

b)

Average profit = P(x)/x = (0.03x² +250x -80)/x = 0.03x +250 -80/x

Marginal profit =P'(x) =0.06x +250

c)

when x = 500

Average profit = 0.03x +250 -80/x =264.84

Marginal profit =0.06(500) +250 =280

d)

The average profit per item for each of the first 500 item produced is $264.84

e)

The proft for 501 item produced and sold is $280

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