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Artist 52 [7]
3 years ago
12

A gardener wants to make a rectangular enclosure using a wall as one side and 120 m of fencing for the other three sides. expres

s the area in terms of x, and find the value of x that gives the greatest area.

Mathematics
1 answer:
Solnce55 [7]3 years ago
3 0
The area is given by A = -2x<span>² + 120x. The</span> greatest area is given by x = 30 m.

Explanation:
See the picture attached for reference.

Let's call:
x = side not facing the wall
We know that the total fence is 120m, therefore
(120 - 2x) = side facing the wall

Note: you could choose to be x = side facing the wall, but the calculations would be a little bit more complicated.

We can now calculate  the area of a rectangle:
A = b · h =
   = x · (120 - 2x)
   = -2x² + 120x

In order to find a maximum for this function, we need to calculate the first derivative:
\frac{d}{dx} (-2x^{2}  + 120x ) = -4x +120

Then, we need to find the candidate points by setting the derivative equal to zero:
<span>-4x +120 = 0
-4(x - 30) = 0
x = 30

Now, in order to understand if the candidate point is a maximum or a minium, let's calculate the second derivative:

</span><span>\frac{d}{dx}(-4x + 120) = -4</span>

According to the "Second Derivative Test", if the second derivative is negative, the point is a local maximum.

Hence, x = 30m gives the greatest area, and we would have:
side not facing the wall = 30m
side facing the wall = 120 - 2·30 = 60m
area = 30 · 60 = 1800 m² 

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