The question is asking for the largest possible area

BUT it only gives you details on the perimeter of the fence. One of the sides of the area enclosure will be used from his fence and he has a maximum of 1,000 meters available for a perimeter... So the first step is to set up a perimeter equation and solve for the dimensions, before solving the max area problem

The equation for the perimeter will be:
2 (Width) + 1 (Length) = 1,000 m
Theres only 1 length being included in the equation because the fence the farmer has will be used instead of the building materials.

Solving for Length =
L = 1,000 m - 2 Width

Now using this equation (and replacing the L) into Area of a rectangle:
Length * Width = (1,000 m - 2 W) * W =

Area = - 2(W^2) + 1,000 W

Now that we have the area, there are a few methods to find the maximum amount. Using calculus we can find the derivative of the function and solve for critical points to find extrema points and find the absolute maximum area.

You can also solve this using algebra by factoring this function because its a quadratic and then you can solve for the highest values. y = -2x^2 +1000x
= -2x(x + 500) where x = 0 and -500

Checking at x = -1, y(-1) = 998

***** EDIT *****

Sorry, the question is only asking for the dimensions of the fence. Its not asking for you to solve the actual maximum size.

The dimensions would be 2 Width x Length

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3 years ago