Question

If 3x+2y=18 find x and y for which the product xy takes its maximum value

Answer 1

Answer:

  (x, y) = (3, 4.5)

Step-by-step explanation:

You want to find x and y such that xy is a maximum subject to the constraint that 3x+2y=18.

Objective function

The objective function f(x, y) = xy is most easily maximized if it can be written in terms of one variable only. Solving the constraint equation for y, we have ...

  3x +2y = 18

  2y = 18 -3x

  y = 1/2(18 -3x) = (3/2)(6 -x)

Then the objective function is ...

  f(x) = x(3/2)(6 -x)

Maximum

We recognize this as the equation of a parabola that opens downward, and has zeros at x=0 and x=6. Its line of symmetry is halfway between those, at x = 3. That is the location of the maximum of the objective function.

The corresponding y-value is ...

  y = (3/2)(6 -3) = 9/2 = 4.5

The values of x and y that maximize their product are (x, y) = (3, 4.5).

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Additional comment

In the attached graph, the larger the product, the farther away the curve is from the origin. The product is maximized subject to the constraint when the curves are tangent to each other. That point of tangency is (x, y) = (3, 4.5).