Questions: 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)=xp(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 function C and price function p. Complete parts (a) through (d). C(x)=-0.02 x^2+50 x+110, p(x)=200, a=500 a. Find the profit function P. The profit function is P(x)=150 x+0.02 x^2-110. b. Find the average profit function and marginal profit function. The average profit function is P(x)/x=150+0.02 x-110/x. The marginal profit function is dP/dx=150+0.04 x. 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 159.78.

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)=xp(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 function C and price function p. Complete parts (a) through (d). C(x)=-0.02 x^2+50 x+110, p(x)=200, a=500 a. Find the profit function P. The profit function is P(x)=150 x+0.02 x^2-110. b. Find the average profit function and marginal profit function. The average profit function is P(x)/x=150+0.02 x-110/x. The marginal profit function is dP/dx=150+0.04 x. 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 159.78.
Transcript text: Let $\mathrm{C}(\mathrm{x})$ represent the cost of producing $x$ items and $p(x)$ be the sale price per item if $x$ items are sold. The profit $\mathrm{P}(\mathrm{x})$ of selling x items is $\mathrm{P}(\mathrm{x})=\mathrm{xp}(\mathrm{x})-\mathrm{C}(\mathrm{x})$ (revenue minus costs). The average profit per item when $x$ items are sold is $\frac{P(x)}{x}$ and the marginal profit is $\frac{d P}{d x}$. The marginal profit approximates the profit obtained by selling one more item given that $x$ items have already been sold. Consider the following cost function C and price function p . Complete parts (a) through (d). \[ C(x)=-0.02 x^{2}+50 x+110, p(x)=200, a=500 \] a. Find the profit function $P$. The profit function is $\mathrm{P}(\mathrm{x})=150 \mathrm{x}+0.02 \mathrm{x}^{2}-110$. b. Find the average profit function and marginal profit function. The average profit function is $\frac{P(x)}{x}=150+0.02 x-\frac{110}{x}$. The marginal profit function is $\frac{d P}{d x}=150+0.04 \mathrm{x}$. c. Find the average profit and marginal profit if $\mathrm{x}=\mathrm{a}$ units have been sold. The average profit if $\mathrm{x}=\mathrm{a}$ units have been sold is $\$ 159.78$.
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Solution

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Solution Steps

Solution Approach

To solve the given problem, we need to:

  1. Define the profit function \( P(x) \) using the given cost function \( C(x) \) and price function \( p(x) \).
  2. Calculate the average profit function by dividing the profit function \( P(x) \) by \( x \).
  3. Determine the marginal profit function by differentiating the profit function \( P(x) \) with respect to \( x \).
  4. Evaluate the average profit and marginal profit at \( x = a \).
Step 1: Profit Function

The profit function \( P(x) \) is defined as the revenue from selling \( x \) items minus the cost of producing \( x \) items. Given the cost function \( C(x) = -0.02x^2 + 50x + 110 \) and the price function \( p(x) = 200 \), we can express the profit function as: \[ P(x) = xp(x) - C(x) = 200x - (-0.02x^2 + 50x + 110) = 0.02x^2 + 150x - 110 \]

Step 2: Average Profit Function

The average profit function is calculated by dividing the profit function by the number of items sold \( x \): \[ \text{Average Profit} = \frac{P(x)}{x} = \frac{0.02x^2 + 150x - 110}{x} = 0.02x + 150 - \frac{110}{x} \]

Step 3: Marginal Profit Function

The marginal profit function is obtained by differentiating the profit function \( P(x) \) with respect to \( x \): \[ \text{Marginal Profit} = \frac{dP}{dx} = 0.04x + 150 \]

Step 4: Evaluate at \( x = a \)

Now, we evaluate both the average profit and marginal profit at \( x = a = 500 \):

  • Average Profit at \( x = 500 \): \[ \text{Average Profit at } x = 500 = 159.78 \]
  • Marginal Profit at \( x = 500 \): \[ \text{Marginal Profit at } x = 500 = 170.00 \]

Final Answer

The average profit when \( x = 500 \) is \( 159.78 \) and the marginal profit is \( 170.00 \). Thus, the answers are: \[ \boxed{\text{Average Profit} = 159.78} \] \[ \boxed{\text{Marginal Profit} = 170.00} \]

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