Answer:
The maximum profit will be $28,800 when 240 acres go for apples and 0 acres go for peaches
Step-by-step explanation:
Let x be the number of acres with apples and y be the numbere of acres with peaches. Note that ![x\ge 0, \ y\ge 0.](https://tex.z-dn.net/?f=x%5Cge%200%2C%20%5C%20y%5Cge%200.)
The grower has 250 acres of land available, then
![x+y\le 250](https://tex.z-dn.net/?f=x%2By%5Cle%20250)
It takes 1 day to fertilize an acre of apples, so it takes x days to fertilize x acres of apples.
It takes 2 days to fertilize 1 acre of peaches, so it takes 2y days to fertilize y acres of peaches.
There are 240 days a year available for fertilizing, so
![x+2y\le 240](https://tex.z-dn.net/?f=x%2B2y%5Cle%20240)
The profit is $120 per acre of apples and $215 per acre of peaches, then the total profit is
![P=120x+215y](https://tex.z-dn.net/?f=P%3D120x%2B215y)
We get the function
which must maximized using restrictions
![\left\{\begin{array}{l}x\ge 0\\y\ge 0\\x+y\le 250\\ x+2y\le 240\end{array}\right.](https://tex.z-dn.net/?f=%5Cleft%5C%7B%5Cbegin%7Barray%7D%7Bl%7Dx%5Cge%200%5C%5Cy%5Cge%200%5C%5Cx%2By%5Cle%20250%5C%5C%20x%2B2y%5Cle%20240%5Cend%7Barray%7D%5Cright.)
Show the solution set of this system of inequalities graphically.
The maximum profit can be at the vertices of this region:
![P(0,120)=120\cdot 0+215\cdot 120=\$25,800\\ \\P(240,0)=120\cdot 240+215\cdot 0=\$28,800](https://tex.z-dn.net/?f=P%280%2C120%29%3D120%5Ccdot%200%2B215%5Ccdot%20120%3D%5C%2425%2C800%5C%5C%20%5C%5CP%28240%2C0%29%3D120%5Ccdot%20240%2B215%5Ccdot%200%3D%5C%2428%2C800)
The maximum profit will be $28,800 when 240 acres go for apples and 0 acres go for peaches