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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 P(x) of selling x items is P(x)=x p(x)-C(x) (revenue minus costs). The average profit per item when x items are sold is P(x)/x and the marginal profit is dP/dx. 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 partsâ (a) throughâ (d) below.
a. Find the profit function P.
b. Find the average profit function and marginal profit function.
c. Find the average profit and marginal profit if x=a units are sold.
d. Interpret the meaning of the values obtained in part (c).
C(x)=â0.02x^2 +50x+100, p(x)=100, a=500

Respuesta :

MrRoyal

Answer:

[tex](a)[/tex] [tex]P(x) = 0.02x^2 +50x - 100[/tex]

[tex](b)[/tex] [tex]Average = 0.02x + 50 - \frac{100}{x}[/tex] and [tex]Marginal = 0.04x + 50[/tex]

[tex](c)[/tex] [tex]Average = 59.8[/tex] and [tex]Marginal = 70[/tex]

(d) See Explanation

Step-by-step explanation:

Given

[tex]p(x) = 100[/tex]

[tex]C(x) = -0.02x^2 +50x +100[/tex]

Solving (a) Profit function; P(x)

[tex]P(x) = xp(x) - C(x)[/tex]

This gives:

[tex]P(x) = x*100 - (-0.02x^2 + 50x + 100)[/tex]

[tex]P(x) = 100x + 0.02x^2 - 50x - 100[/tex]

Collect like terms

[tex]P(x) = 0.02x^2 - 50x +100x - 100[/tex]

[tex]P(x) = 0.02x^2 +50x - 100[/tex]

Solving (b): Average profit function and Marginal profit function

[tex]Average = \frac{P(x)}{x}[/tex]

This gives:

[tex]Average = \frac{0.02x^2 + 50x - 100}{x}[/tex]

Break down the fraction

[tex]Average = \frac{0.02x^2}{x} + \frac{50x}{x} - \frac{100}{x}[/tex]

[tex]Average = 0.02x + 50 - \frac{100}{x}[/tex]

[tex]Marginal = \frac{dP}{dx}[/tex]

[tex]P(x) = 0.02x^2 +50x - 100[/tex]

Differentiate

[tex]\frac{dP}{dx} = 2 * 0.02x + 50 - 0[/tex]

[tex]\frac{dP}{dx} = 0.04x + 50[/tex]

Hence:

[tex]Marginal = 0.04x + 50[/tex]

Solving (c): Average profit and Marginal profit if x = a

[tex]a = 500[/tex]

So:

[tex]x =500[/tex]

Substitute 500 for x

[tex]Average = 0.02x + 50 - \frac{100}{x}[/tex]

[tex]Average = 0.02 * 500 + 50 - \frac{100}{500}[/tex]

[tex]Average = 59.8[/tex]

[tex]Marginal = 0.04x + 50[/tex]

[tex]Marginal = 0.04*500 + 50[/tex]

[tex]Marginal = 70[/tex]

Solving (d): Interpret the values in (c)

[tex]Average = 59.8[/tex]

They make a profit of 59.8 for the first 500 items

[tex]Marginal = 70[/tex]

From the 501st item, the profit is 70

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