SOLUTION: The sum of two numbers is 64. Show that their product has a maximum of 1024. Many thanks

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: The sum of two numbers is 64. Show that their product has a maximum of 1024. Many thanks      Log On


   

Question 895796: The sum of two numbers is 64. Show that their product has a maximum of 1024. Many thanks
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
x%2By=64. xy%3C=1024 ?

y=64-x.
x%2864-x%29=f-----the product of x and y.
f=64x-x%5E2
f=-x%5E2%2B64x

Complete The Square. Use %2864%2F2%29%5E2=32%5E2.
f=-1%28x%5E2-64x%29
f=-1%28x%5E2-64x%2B32%5E2-32%5E2%29
f=-1%28%28x-32%29%5E2-32%5E2%29
highlight_green%28f=-%28x-32%29%5E2%2B32%5E2%29 ------ standard form, parabolic function.

The maximum, the vertex of f is at (32,32^2) and 32%5E2=1024.


(See my lesson on Completing The Square)