Maximum area

Geometry Level 1

A rectangle has integer sides and a perimeter of 90. What is the largest possible area of the rectangle?

484 506 529 552

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Arron Kau by Arron Kau

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5 solutions

Jiunn Shan Tan
Aug 6, 2013

2X + 2Y = 90 ; AREA = XY

A = X(45-X)

dA/dX = 45 - 2x = 0

X = 22.5 = 23 ; Y = 22

XY = (22)(23) = 506

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Victor Carnaúba
Aug 4, 2013

possible sides are closest 22x23=506

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Let x and y be the length and width of the rectangle. 2x + 2y = 90, hence x + y = 45.

By AM-GM, the area, xy ≤ ( 45 2 ) 2 \le \left(\frac{45}{2} \right)^2 → \rightarrow xy ≤ 506.25 \le 506.25

Therefore the largest possible area of the rectangle is 506.

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Moderator note:

While this solution looks good, it is actually incorrect. When you solve an inequality for all real numbers, solving it for the integers doesn't necessarily mean that you take the next integer value.

For example: If x x and y y are positive integers such that 7 x + 9 y ≤ 22 7x + 9y \leq 22 , what is the maximum value of x + y x+ y ? If we tried to do this over the positive reals, we can get x + y < 22 7 x + y < \frac{22}{7} . The next smallest integer is 3. However, the answer to this question over the positive integers is 2, corresponding to ( x , y ) = ( 1 , 1 ) (x,y) = (1,1) .

2x + 2y = 90 --> 2 (x + y) = 90 --> x + y = 45 --> x= 23 e y=22 --> 23 * 22 = 506

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Edsel Salariosa
Aug 4, 2013

Using maxima and minima of calculus.First equation is the perimeter of rectangle P=2l+2w. The 2nd equation is A=LxW for the area of rectangle.We have 90=2l+2w for the perimeter.Using algebra we have 45-L=w for the 3rd equation.Substitute the 2nd equation to the3rd equation ,them we have A=L(45-L).Using calculuus we have DA/DL=45-2l. DA/DL is equal to 0. We get L=22.5 and w=22.5.Area =22.5^2

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Moderator note:

You cannot apply calculus to a discrete set. Just knowing the maximum over the reals does not tell you about the maximum over the integers.

This is one reason why solving linear inequalities over the integers is much harder than solving linear inequalities over the real numbers.

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