Answer
(a) See graph.
\begin{cases} x+y\leq300\hspace2cm (land-total) \\ 70x+35y\leq17500\hspace1cm (planting- budget)\\25x+55y\leq12000\hspace1cm (fertilizer- budget)\\x\geq0,y\geq0 \end{cases}
(b) Yes.
(c) No.
Work Step by Step
(a) Assume the farmer can plant $x$ acres of cauliflower and $y$ acres of cabbage, we can set up the following inequalities:
\begin{cases} x+y\leq300\hspace2cm (land-total) \\ 70x+35y\leq17500\hspace1cm (planting- budget)\\25x+55y\leq12000\hspace1cm (fertilizer- budget)\\x\geq0,y\geq0 \end{cases}
We can graph the inequalities with the feasible region (solution region).
(b) To check if the farmer can plant 155 acres of cauliflower and 115 acres of cabbage, we locate the coordinate $(155,115)$ and see if the point fall in the solution region, and the answer is Yes.
(c) To check if the farmer can plant 115 acres of cauliflower and 175 acres of cabbage, we locate the coordinate $(115,175)$ and see if the point fall in the solution region, it appears to fall on the purple boarder line. Plug the coordinates into the third equation above, we have $25\times115+55\times175=12500\gt12000$, so the answer is NO.
