question 5\nyou and your friends want to start your own business doing yard - work, where you will clean…

question 5\nyou and your friends want to start your own business doing yard - work, where you will clean yards the fargo - moorhead area. after some research, you estimate that the total cost of cleaning y yards is given by the function.\nc(y)=5.50y + 1000\nwhen the unit price to clean one yard is p dollars per yard, you anticipate a demand of y yards to be cleaned, according to the demand equation\n8p+5y = 400\ngiven this information, what is the maximum amount of profit that can be made cleaning yards?\nround your answer appropriately and input just the number that you get. do not include any labels or units.

question 5\nyou and your friends want to start your own business doing yard - work, where you will clean yards the fargo - moorhead area. after some research, you estimate that the total cost of cleaning y yards is given by the function.\nc(y)=5.50y + 1000\nwhen the unit price to clean one yard is p dollars per yard, you anticipate a demand of y yards to be cleaned, according to the demand equation\n8p+5y = 400\ngiven this information, what is the maximum amount of profit that can be made cleaning yards?\nround your answer appropriately and input just the number that you get. do not include any labels or units.

Answer

Explanation:

Step1: Express price $p$ in terms of $Y$

From $8p + 5Y=400$, we get $p=\frac{400 - 5Y}{8}=50-\frac{5Y}{8}$.

Step2: Calculate revenue function $R(Y)$

Revenue $R(Y)=p\times Y=(50 - \frac{5Y}{8})Y = 50Y-\frac{5Y^{2}}{8}$.

Step3: Calculate profit function $\pi(Y)$

Profit $\pi(Y)=R(Y)-C(Y)$. Given $C(Y)=5.5Y + 1000$, then $\pi(Y)=50Y-\frac{5Y^{2}}{8}-(5.5Y + 1000)=-\frac{5Y^{2}}{8}+44.5Y - 1000$.

Step4: Find the derivative of the profit function

$\pi^\prime(Y)=-\frac{5\times2Y}{8}+44.5=-\frac{5Y}{4}+44.5$.

Step5: Set the derivative equal to zero to find critical - points

$-\frac{5Y}{4}+44.5 = 0$. Then $\frac{5Y}{4}=44.5$, and $Y=\frac{44.5\times4}{5}=35.6$.

Step6: Find the second - derivative of the profit function

$\pi^{\prime\prime}(Y)=-\frac{5}{4}<0$, so when $Y = 35.6$, the profit is maximized.

Step7: Calculate the maximum profit

Substitute $Y = 35.6$ into the profit function $\pi(Y)$. $\pi(35.6)=-\frac{5\times(35.6)^{2}}{8}+44.5\times35.6 - 1000$ $=-\frac{5\times1267.36}{8}+1588.2 - 1000$ $=-792.1+1588.2 - 1000$ $=-203.9$.

Answer:

$- 203.9$