Question

A farmer plans to fence a rectangular garden next to a river. There will be no fence along the river side. Draw a picture of this situation. The pasture will contain 405,000 square meters. Find the dimensions of the garden that will require the least amount of fencing.

Answer 1

x = 450 m

y = 900 m

P (min) = 1800 m

Explanation:

Let´s call "x" and "y" de sides of the rectangular garden, y is the side parallel to the river ( that is it will be fenced only one )

Then:

2*x + y = P ( perimeter of the area)

And x * y = 405000 m²

y = 405000/ x

And P as a function of x is:

P(x) = 2*x + 405000/x

Tacking derivatives relative to x on both sides of the equation.

P´(x) = 2 - 405000/x²

P´(x) = 0 2*x² - 405000 = 0

x² = 202500

x = 450 m

And y = 405000 / 450

y = 900 m

And

P(min) = 2*450 + 900

P(min) = 1800 m

We know P is minimm at x =450 snce the second derivative of P

P´´(x) = 405000*2*x / x⁴ is always positive