Question

A farmer is going to divide her40 acre farm between two crops. Seed for crop A costs $30 per acre. Seedfor crop B costs $15 per acre. The farmer can spend at most $1050 on seed.If crop B brings in a profit of $70 per acre, and crop A brings in a profit of $180 per acre, how many acres ofeach crop should the farmer plant to maximize her profit?acres of crop Aacres of crop B

Answer 1

Given: A farmer is going to divide her40 acre farm between two crops

To Determine: How many acres of each crop should the farmer plant to maximize her profit

Solution

`\begin{gathered} x=acres\text{ for crop A} \\ y=acres\text{ fro corp B} \end{gathered}`
`\begin{gathered} Therefore: \\ x+y\leq40 \\ 30x+15y\leq1050 \\ Profit\text{ functon is} \\ P=70y+180x \end{gathered}`

The graph of the inequalitiesis as shown below

Neglecting the negative axis, the point that will maximize the profit are te points shown above with their coordinates

The coordinates will maximum profits are

`\begin{gathered} Point1:(0,40) \\ Point2:(35,0) \\ Point3:(30,10) \end{gathered}`

Substitute the 3 points into the profit functions to determine thepoint that will given the maximum profit

`\begin{gathered} P=180x+70y \\ Point1 \\ P1=180(0)+70(40) \\ P1=0+2800 \end{gathered}`
`\begin{gathered} Point2 \\ P2=180(35)+70(0) \\ P2=6300+0 \\ P2=6300 \end{gathered}`
`\begin{gathered} Point3 \\ P3=180(30)+70(10) \\ P3=5400+700 \\ P3=6100 \end{gathered}`

Hence, the point that will maximize profit is (35, 0), which is

35 acres of crop A, and

0 acre of crop B