The maximum area of a shape is the highest area, the shape can have.
The dimension of the rectangle that will produce the maximum area is 4 units by 2 units
The equation is given as:
[tex]\mathbf{x + 2y - 8 = 0}[/tex]
Rewrite as:
[tex]\mathbf{2y= 8- x}[/tex]
Divide both sides by 2
[tex]\mathbf{y= \frac{8- x}{2}}[/tex]
[tex]\mathbf{y= 4- \frac{x}{2}}[/tex]
The area (A) of the rectangle is:
[tex]\mathbf{A = xy}[/tex]
Substitute [tex]\mathbf{y= 4- \frac{x}{2}}[/tex]
[tex]\mathbf{A = x(4- \frac x2)}[/tex]
Expand
[tex]\mathbf{A = 4x- \frac{x^2}{2}}[/tex]
Differentiate
[tex]\mathbf{A' = 4- x}[/tex]
Set the derivative to 0
[tex]\mathbf{4- x = 0}[/tex]
Collect like terms
[tex]\mathbf{x = 4}[/tex]
Recall that: [tex]\mathbf{y= 4- \frac{x}{2}}[/tex]
Substitute 4 for x
[tex]\mathbf{y = 4- \frac{4}{2}}[/tex]
[tex]\mathbf{y = 4- 2}[/tex]
[tex]\mathbf{y = 2}[/tex]
So, we have:
[tex]\mathbf{x = 4}[/tex] and [tex]\mathbf{y = 2}[/tex]
Hence, the dimension of the rectangle that will produce the maximum area is 4 units by 2 units
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