Let's break down each part of the question and solve it step-by-step.
### Part (A): Calculating the Average Profit per Mower if 30 Mowers are Produced
To find the average profit per mower when 30 mowers are produced, we first need to calculate the total profit, [tex]\( P(30) \)[/tex], and then divide it by the number of mowers, which is 30.
The profit function is given by:
[tex]\[ P(x) = 80x - 0.08x^2 - 550 \][/tex]
First, we calculate [tex]\( P(30) \)[/tex]:
[tex]\[ P(30) = 80(30) - 0.08(30)^2 - 550 \][/tex]
From the result, we know:
[tex]\[ P(30) = 1778 \, \text{dollars} \][/tex]
Now, the average profit per mower is:
[tex]\[ \text{Average profit per mower} = \frac{P(30)}{30} \][/tex]
[tex]\[ \text{Average profit per mower} = \frac{1778}{30} \][/tex]
[tex]\[ \text{Average profit per mower} = 59.27 \, \text{dollars (rounded to 2 decimal places)} \][/tex]
### Part (B): Calculating the Marginal Average Profit at a Production Level of 30 Mowers
The marginal average profit is the change in the total profit when one additional mower is produced.
First, calculate [tex]\( P(31) \)[/tex]:
[tex]\[ P(31) = 80(31) - 0.08(31)^2 - 550 \][/tex]
[tex]\[ P(31) = 1853.12 \, \text{dollars} \][/tex]
Now, the marginal profit when increasing production from 30 to 31 mowers is:
[tex]\[ \text{Marginal profit} = P(31) - P(30) \][/tex]
[tex]\[ \text{Marginal profit} = 1853.12 - 1778 \][/tex]
[tex]\[ \text{Marginal profit} = 75.12 \, \text{dollars (rounded to 2 decimal places)} \][/tex]
### Part (C): Estimating the Average Profit per Mower if 31 Mowers are Produced
We can now estimate the average profit per mower if 31 mowers are produced, using:
[tex]\[ \text{Average profit per mower} = \frac{\text{Total profit}}{\text{Number of mowers}} \][/tex]
Here, the total profit for 31 mowers can be estimated by adding the marginal profit for the 31st mower to the total profit for 30 mowers:
[tex]\[ \text{Estimated total profit} = P(30) + \text{Marginal profit} \][/tex]
[tex]\[ \text{Estimated total profit} = 1778 + 75.12 \][/tex]
[tex]\[ \text{Estimated total profit} = 1853.12 \, \text{dollars} \][/tex]
Finally, the estimated average profit per mower for 31 mowers is:
[tex]\[ \text{Estimated average profit per mower} = \frac{1853.12}{31} \][/tex]
[tex]\[ \text{Estimated average profit per mower} = 59.78 \, \text{dollars (rounded to 2 decimal places)} \][/tex]
### Summary of Results:
- (A) The average profit per mower when 30 mowers are produced is \[tex]$59.27.
- (B) The marginal average profit at a production level of 30 mowers is \$[/tex]75.12.
- (C) The estimated average profit per mower if 31 mowers are produced is \$59.78.
