To find the maximum for the profit function [tex]\( P = 3x + 12y \)[/tex] subject to the constraints:
[tex]\[
\left\{\begin{array}{l}
x + y \leq 80 \\
x - 2y \geq 15 \\
x \geq 0 \\
y \geq 0
\end{array}\right.
\][/tex]
we need to determine the feasible region first. The constraints define a feasible region in the first quadrant, and the potential solutions for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] will correspond to the vertices (corners) of this region. We will examine these corners and then evaluate the profit function at each to find the one that returns the maximum value.
Here is how we proceed step by step:
1. Identify the constraints:
- [tex]\( x + y \leq 80 \)[/tex]
- [tex]\( x - 2y \geq 15 \)[/tex]
- [tex]\( x \geq 0 \)[/tex]
- [tex]\( y \geq 0 \)[/tex]
2. Convert the inequalities:
- Rewrite [tex]\( x - 2y \geq 15 \)[/tex] as [tex]\( x \geq 15 + 2y \)[/tex].
3. Determine the intersection points of the constraints:
- Intersection of [tex]\( x + y = 80 \)[/tex] and [tex]\( x - 2y = 15 \)[/tex]: Solve simultaneously:
[tex]\[
\begin{cases}
x + y = 80 \\
x - 2y = 15
\end{cases}
\][/tex]
Adding these equations, we get:
[tex]\[
(x + y) + (x - 2y) = 80 + 15 \implies 2x - y = 95
\][/tex]
Simplifying to find:
[tex]\[
2x = 95 + y \implies x = 47.5 + 0.5y
\][/tex]
Substituting [tex]\( x = 47.5 + 0.5y \)[/tex] back into [tex]\( x + y = 80 \)[/tex]:
[tex]\[
47.5 + 0.5y + y = 80 \implies 47.5 + 1.5y = 80 \implies 1.5y = 32.5 \implies y \approx 21.67
\][/tex]
Then, substituting [tex]\( y \)[/tex] back to find [tex]\( x \)[/tex]:
[tex]\[
x + 21.67 = 80 \implies x \approx 58.33
\][/tex]
4. Check the endpoints and evaluate the profit function:
- At [tex]\( (x, y) = (58.33, 21.67) \)[/tex], compute:
[tex]\[
P = 3(58.33) + 12(21.67) \implies P = 175 + 260 \approx 435.0
\][/tex]
5. Summarize the result:
- The values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that maximize the profit function, given the constraints, are approximately [tex]\( x = 58.33 \)[/tex] and [tex]\( y = 21.67 \)[/tex].
- The maximum profit, at these points, is approximately [tex]\( P = 435.0 \)[/tex].
Therefore, the maximum profit is [tex]\(\$435.00\)[/tex], with [tex]\(x = 58.33\)[/tex] and [tex]\(y = 21.67\)[/tex].
