Solving for maximum area?

Question

You have 188 feet of fencing to enclose a rectangular region. What is the maximum area?

Answer 1

It is known that the maximum area for a rectanglular figure is found when the shape is (specifically) a square.

To find the length of a side of the square:
188/4 = 47

A square of 47 feet per side will produce the max area.

A = 47² = 2209 sq ft

Answer 2

Ofcourse you have enough info. Let's call the sides of the rectangle a and b. Then:

circumference = 2*a + 2*b = 188

area = a*b

By using the first equation you can express b in terms of a:

a = 94-b

Then the area becomes:

area = (94-b)*b = 94*b-b^2

You can now find the value of b for which the area is maximum by differentiating the area with respect to b and equating this to zero:

darea/db = 94-2*b = 0

=> b = 47

If b = 47 feet, then a = 47 feet as well. The optimal shape seems to be a square.

Conclusion: the maximum area is 47*47 = 2209 ft.

Answer 3

Call the sides x and y.

Maximize area: A = xy

Perimeter: 188 = 2x + 2y
Divide by 2: 94 = x + y
Subtract y: 94 - y = x

Substituting: A = y(94-y) = 94y - y^2

A' = 94 - 2y

Set A' = 0, solve: y = 94/2 = 47
Then x = 94-47 = 47

Area = 47*47 = 2209

This is how to show work using Calculus. If you are only in Algebra, go back to the part where
A = 94y - y^2

Vertex of parabola = -b/2a = -94/-2 = 47 = y, then find x = 47 as shown in above solution to get same result.

Answer 4

It is known that a square will provide you with the maximum area for a rectangle but if you didn't know this, you could write this equation:

P = 2x + 2y = 188 (1)
A = X * Y (2)

y = 94 - x (3)(from 1)
A = X * (94-X)
= 94x - x^2

To find a maximum, take the derivative and solve for 0

dA/dx = 94 -2x = 0
94 = 2x
47 = x

y = 94 - x = 47

So x = y and it is indeed a square with sides of length 47.

Answer 5

Since there is not wall or river or something else to edge up against, it will always turn out to be a square.

Lets use calc.

If the length = x, then the width = 94 - x.

The area = x(94 - x) = 94x - x^2

Take the derivative: area' = 94 - 2x.

Set this = 0 to get the vertex of the downward opening parabola. 94 = 2x, x = 47.

l = 47, w = 47. Its a square.

Answer 6

Let an edge be x and the other one be (188-2x)/2 = 94-x

The area is:

A(x) = x*(94-x) = -x^2 + 94x

You can find the maximum value of that function equalling the first derivative to zero and finding the roots:

A'(x) = -2x+94

-2x+94 = 0
x = 94/2 = 47

The maximum area is:

A(99/2) = -(47)^2 + 47*94= 2209 ft^2

Answer 7

a times b =188
solve for a a=188-b

Answer 8

agreed not enough info

edit: ohhh i didnt realize that

Answer 9

not enough info.