Answer :
Sure! Let's walk through the solution step by step:
(a) Finding the Total Profit from Manufacturing and Selling 200 Units:
1. Understand the Marginal Profit Function:
The marginal profit function is given as [tex]\( P'(x) = -0.0006x^2 + 0.08x + 24 \)[/tex]. This represents the rate of change of the profit for each additional unit manufactured and sold.
2. Calculate the Total Profit from 0 to 200 Units:
To find the total profit when manufacturing and selling 200 units, we need to integrate the marginal profit function from 0 to 200. This will give us the change in profit over that interval.
3. Add the Initial Profit:
The initial profit is given as [tex]$800, which is the profit when 0 units are produced. So, the total profit after producing and selling 200 units is the integrated result plus this initial profit.
The total profit realizable from the manufacture and sale of 200 units per day is $[/tex]5,600.
(b) Additional Daily Profit from Increasing Production from 200 to 220 Units:
1. Integrate the Marginal Profit Function from 200 to 220:
To find the additional profit from increasing production from 200 to 220 units, integrate the marginal profit function over this range.
2. Evaluate the Increase in Profit:
The result of this integration gives you the extra profit obtained by producing and selling the additional 20 units.
The additional daily profit realized by increasing production from 200 to 220 units is approximately $286.40.
I hope this helps! Let me know if you have any questions about the process.
(a) Finding the Total Profit from Manufacturing and Selling 200 Units:
1. Understand the Marginal Profit Function:
The marginal profit function is given as [tex]\( P'(x) = -0.0006x^2 + 0.08x + 24 \)[/tex]. This represents the rate of change of the profit for each additional unit manufactured and sold.
2. Calculate the Total Profit from 0 to 200 Units:
To find the total profit when manufacturing and selling 200 units, we need to integrate the marginal profit function from 0 to 200. This will give us the change in profit over that interval.
3. Add the Initial Profit:
The initial profit is given as [tex]$800, which is the profit when 0 units are produced. So, the total profit after producing and selling 200 units is the integrated result plus this initial profit.
The total profit realizable from the manufacture and sale of 200 units per day is $[/tex]5,600.
(b) Additional Daily Profit from Increasing Production from 200 to 220 Units:
1. Integrate the Marginal Profit Function from 200 to 220:
To find the additional profit from increasing production from 200 to 220 units, integrate the marginal profit function over this range.
2. Evaluate the Increase in Profit:
The result of this integration gives you the extra profit obtained by producing and selling the additional 20 units.
The additional daily profit realized by increasing production from 200 to 220 units is approximately $286.40.
I hope this helps! Let me know if you have any questions about the process.
