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34kurt
2 years ago
11

A rectangular area is formed having a perimeter of 40 cm. Determine the length and breadth of the rectangle if it is to enclose

the maximum possible area​
Mathematics
1 answer:
Vilka [71]2 years ago
5 0

Let x and y be the dimensions of the rectangle. If the perimeter is 40, we have

2(x+y)=40 \iff x+y=20

We can expression one variable in terms of the others as

x+y=20 \iff x=20-y

Since the area is the product of the dimensions, we have

xy=(20-y)y=-y^2+20y

This is a parabola facing down, so it's vertex is the maximum:

f(y)=-y^2+20y \implies f'(y)=-2y+20

So, the maximum is

f'(y)=0 \iff -2y+20=0 \iff 2y=20 \iff y=10

And since we know that x+y=20, we have x=10 as well.

This is actually a well known theorem: out of all the rectangles with given perimeter, the one with the greatest area is the square.

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