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Question
The maximum value of z = 5x + 3y subject to the constraints 3x + 5y ≤ 15, 5x + 2y ≤ 10, x, y ≥ 0 is ______.
Options
235
`235/9`
`235/19`
`235/3`
Solution
The maximum value of z = 5x + 3y subject to the constraints 3x + 5y ≤ 15, 5x + 2y ≤ 10, x, y ≥ 0 is `bb(underline(235/19)`.
Explanation:
Step 1: Find the critical points of the given function.
In the question, a function z = 5x + 3y is given, and the constraints 3x + 5y ≤ 15, 5x + 2y ≤ 10, x, y ≥ 0 is also given.
Draw a graph describing the given inequalities as follows:
From the graph, it is clear that the critical points are (0, 0), (2, 0), (0, 3) and `(20/19, 45/19)`.
Step 2: Find the maximum value of the given function.
Since, the critical points are (0, 0), (2, 0), (0, 3) and `(20/19, 45/19)`.
Evaluate z for (0, 0) as follows:
z = 5(0) + 3(0)
z = 0
So, the value of z for (0, 0) is 0.
Similarly, Evaluate z for (2, 0) as follows:
z = 5(2) + 3(0)
z = 10
So, the value of z for (2, 0) is 10.
Similarly, Evaluate z for (0, 3) as follows:
z = 5(0) + 3(3)
z = 9
So, the value of z for (0, 3) is 9.
Similarly, Evaluate for `(20/19, 45/19)` as follows:
z = `5(20/19) + 3(45/19)`
z = `325/19`
So, the value of z for `(20/19, 45/19)` is `235/19`.
Therefore, the maximum value of the given function is `235/19`.
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