Question

Calculate the largest possible area of the field ?

Answer 1

The maximum area is `710` square units (the question is unclear whether these are feet, meters, yards, etc. )

Explanation:

Consider the following image, which pictures `4x^2 + 9y^2 = 3600`.

Let's start by solving for `y` for the ellipse.

`9y^2 = 3600 - 4x^2`

`y^2 = 400 - 4/9x^2`

`y = +- sqrt(400 - 4/9x^2)`

The area of the inscribed rectangle would be given by `A = 2x(2sqrt(400 - 4/9x^2))= 4xsqrt(400 - 4/9x^2)`

The first derivative of this is

`A' = 4sqrt(400 - 4/9x^2) + (4x(-8/9x^2))/(2sqrt(400 - 4/9x^2))`

`A' = (8(400 - 4/9x^2) - 32/9x^3)/(2sqrt(400 - 4/9x^2)`

Critical points of this will be

`0 = 3200 - 32/9x^2 - 32/9x^3`

`0 = 28800 - 32x^2 - 32x^3`

`0 = 900 - x^2 - x^3`

Solve using a calculator to get `x = 9.333`.

This means the maximum are will be `A = 4(9.333)sqrt(400 - 4/9(9.333)^2) =710` square units (whatever these may be).

Hopefully this helps!