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`

Answer 1

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`.