Question

John wants to fence a rectangular garden, which is 30 feet. What would be the dimensions of the fence enclosing the greatest area?

Answer 1

Step-by-step explanation:

Therefore, the rectangle with perimeter 30 ft which has largest area is the square with side length 7.5 ft. For a given perimeter the rectangle with the greatest area is the square. The garden should be 7.5 ft x 7.5 ft.

Answer

7.5×7.5

algebra

Here is an algebraic solution:

Let x = length and y = width

Then, 2x+2y = 30. So, y = 15 - x

Area = A = xy = x(15 - x)

A = -x2 + 15x

The graph of the area function is a parabola opening downward. The maximum occurs at the vertex, which has x-coordinate x = -15/[2(-1)].

= 7.5

So, the maximum area occurs when x = 7.5 and y = 15 - x = 7.5.

Therefore, the rectangle with perimeter 30 ft which has largest area is the square with side length 7.5 ft.

Answer 2

Answer:

Here is an algebraic solution:

 

Let x = length and y = width

 

Then, 2x+2y = 30.  So, y = 15 - x

 

Area = A = xy = x(15 - x)

 

           A = -x2 + 15x

 

The graph of the area function is a parabola opening downward.  The maximum occurs at the vertex, which has x-coordinate x = -15/[2(-1)].

So, the maximum area occurs when x = 7.5 and y = 15 - x = 7.5.

 

Therefore, the rectangle with perimeter 30 ft which has largest area is the square with side length 7.5 ft.

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