Area of rectangle = 16 square feet
Least amount of material = ?
x = 4 ft and y = 4 ft
We know that a rectangle has area = xy and perimeter = 2x + 2y
We want to use least amount of material to design the sandbox which means we want to minimize the perimeter which can be done by taking the derivative of perimeter and then setting it equal to 0.
So we have
xy = 16
y = 16/x
p = 2x + 2y
put the value of y into the equation of perimeter
p = 2x + 2(16/x)
p = 2x + 32/x
Take derivative with respect to x
d/dt (2x + 32/x)
2 - 32/x²
set the derivative equal to zero to minimize the perimeter
2 - 32/x² = 0
32/x² = 2
x² = 32/2
x² = 16
x = ft
put the value of x into equation xy = 16
(4)y = 16
y = 16/4
y = 4 ft
So the dimensions are x = 4 ft and y = 4 ft in order to use least amount of material.
xy = 16
4*4 = 16
16 = 16 (satisfied)
The least amount of dimensions would be using a 4ft x 4ft base.
When looking to maximize area and minimize materials, a perfect square always does this. To prove the point, consider the 4 x 4 box in the answer to say a 16 x 1 box, which would have the same area.
The 4 x 4 box would have 4 sides each of 4 ft. Therefore, it would have 16 feet of sides to construct.
Meanwhile, there would be two sides on the other one that were 16 ft alone.
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