Answer:
1. Express the width of the rectangle in terms of the length X.
width = 500 - X
2. Express the surface area of the rectangle in terms of X.
area = -X² + 500X
3. What value of X gives the maximum surface area?
maximum surface area results from the rectangle being a square, so 1,000 ÷ 4 = 250
X = 250 ft
4. What is the maximum surface area?
maximum surface area = X² = 250² = 62,500 ft²
Step-by-step explanation:
since the perimeter = 1,000
1,000 = 2X + 2W
500 = X + W
W = 500 - X
area = X · W = X · (500 - X) = 500X - X² or -X² + 500X
The area of a shape is the amount of space it occupies.
The length is represented as x.
Let the width be y.
So, we have:
[tex]\mathbf{Perimeter =2(x + y)}[/tex]
This gives
[tex]\mathbf{2(x + y) = 1000}[/tex]
Divide both sides by 2
[tex]\mathbf{x + y = 500}[/tex]
Make y the subject
[tex]\mathbf{y = 500 -x}[/tex]
So, the width in terms of x is 500 - x
The surface area is calculated as:
[tex]\mathbf{A = xy}[/tex]
Substitute [tex]\mathbf{y = 500 -x}[/tex]
[tex]\mathbf{A = x(500 - x)}[/tex]
So, the surface area in terms of x is x(500 - x)
Expand [tex]\mathbf{A = x(500 - x)}[/tex]
[tex]\mathbf{A = 500x - x^2}[/tex]
Differentiate
[tex]\mathbf{A' = 500- 2x}[/tex]
Equate to 0
[tex]\mathbf{500- 2x = 0}[/tex]
Rewrite as:
[tex]\mathbf{2x = 500}[/tex]
Divide both sides by 2
[tex]\mathbf{x = 250}[/tex]
So, the value of x that gives maximum surface area is 250
Substitute 250 for x in [tex]\mathbf{A = x(500 - x)}[/tex]
[tex]\mathbf{A = 250 \times (500 - 250)}[/tex]
[tex]\mathbf{A = 250 \times 250}[/tex]
[tex]\mathbf{A = 62500}[/tex]
Hence, the maximum area is 62500
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