Questions: The total profit (in dollars) from the sale of x charcoal grills is P(x) = 70x - 0.2x^2 - 290 (A) Find the average profit per grill if 40 grills are produced. Ave. profit = 54.75 Find the marginal average profit at a production level of 40 grills. (B) Marginal average profit = Use the results from parts (A) and (B) to estimate the average profit per grill if 41 grills are produced. (C) Estimated average profit =

The total profit (in dollars) from the sale of x charcoal grills is P(x) = 70x - 0.2x^2 - 290 (A) Find the average profit per grill if 40 grills are produced. Ave. profit = 54.75 Find the marginal average profit at a production level of 40 grills. (B) Marginal average profit = Use the results from parts (A) and (B) to estimate the average profit per grill if 41 grills are produced. (C) Estimated average profit =
Transcript text: (1 point) The total profit (in dollars) from the sale of $x$ charcoal grills is \[ P(x)=70 x-0.2 x^{2}-290 \] (A) Find the average profit per grill if 40 grills are produced. Ave. profit = 54.75 $\square$ Find the marginal average profit at a production level of 40 grills. (B) Marginal average profit $=$ $\square$ Use the results from parts $(A)$ and $(B)$ to estimate the average profit per grill if 41 grills are produced. (C) Estimated average profit $=$ $\square$
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Solution

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Solution Steps

Step 1: Average Profit Calculation

The total profit for selling 40 units is calculated by the formula $P(x) = ax - bx^2 - c$, which gives $P(40) = 70_40 - 0.2_40^2 - 290 = 2190$. The average profit per unit for 40 units is $\frac{P(40)}{40} = 54.75$.

Step 2: Marginal Average Profit Calculation

The marginal average profit at 40 units is the derivative of the average profit function with respect to 40, which gives $46.75$.

Step 3: Estimating Average Profit for x+1 Units

By adding the marginal average profit to the average profit for 40 units, we estimate the average profit per unit for 41 units to be approximately $101.5$.

Final Answer:

The average profit per unit for 40 units is $54.75$. The marginal average profit at this production level is $46.75$. Therefore, the estimated average profit per unit if the production level is increased to 41 units is approximately $101.5$.

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