# Largest Rectangle Under the curve y=-.0064x^2+36?

What is the largest rectangle you can fit under this equation? the rectangle is bound by the equation and the x-axis

Relevance

I assume this is for a calculus class.

given:

y = -0.0064x² + 36

for (x, y) > 0

so

y = height of rectangle

2x = width of rectangle

see

http://i584.photobucket.com/albums/ss282/JimPfoss/...

so

A(x) = area of rectangle

A(x) = 2x(-0.0064x^2 + 36)

= -0.0128x³ + 72x

max area is where A'(x) = 0, so:

A'(x) = -0.0384x² + 72

0.0384x² = 72

x² = 1875

x = 25√(3)

so

width = 50√(3) ≈ 86.6

height = 24

area = 1200 √(3) ≈ 2079 units²

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