SOLUTION: Find two real numbers x and y such that 2x+y=80 and A=xy is maximum

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Question 1082166: Find two real numbers x and y such that 2x+y=80 and A=xy is maximum
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
A=x%2880-2x%29

A=80x-2x%5E2
Maximum at 80-4x=0;
4x=80
x=20 and y=40

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
Find two real numbers x and y such that 2x+y=80 and A=xy is maximum
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From 2x + y = 80 you have y = 80-2x.


Then A = xy = x*(80-2x), and you need to find the maximum of this quadratic function.


The zeros are at x = 0 and x = 80%2F2 = 40.

Hence, the maximum is achieved at x = 40%2F2 = 20, exactly at the mid-point between the zeroes.


Then the maximum value of A is A = x*(80-2x) = 20*(80-2*20) = 20*40 = 800.

Solved.




Plot y = x*(80-2x)


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My other lessons in this site on finding the maximum/minimum of a quadratic function are
    - HOW TO complete the square to find the minimum/maximum of a quadratic function
    - Briefly on finding the minimum/maximum of a quadratic function
    - HOW TO complete the square to find the vertex of a parabola
    - Briefly on finding the vertex of a parabola
    - A rectangle with a given perimeter which has the maximal area is a square
    - A farmer planning to fence a rectangular garden to enclose the maximal area
    - A farmer planning to fence a rectangular area along the river to enclose the maximal area
    - A rancher planning to fence two adjacent rectangular corrals to enclose the maximal area


Also, you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this textbook under the topic "Finding minimum/maximum of quadratic functions".