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:
\(\mathbf{x + 2y - 8 = 0}\)
Rewrite as:
\(\mathbf{2y= 8- x}\)
Divide both sides by 2
\(\mathbf{y= \frac{8- x}{2}}\)
\(\mathbf{y= 4- \frac{x}{2}}\)
The area (A) of the rectangle is:
\(\mathbf{A = xy}\)
Substitute \(\mathbf{y= 4- \frac{x}{2}}\)
\(\mathbf{A = x(4- \frac x2)}\)
Expand
\(\mathbf{A = 4x- \frac{x^2}{2}}\)
Differentiate
\(\mathbf{A' = 4- x}\)
Set the derivative to 0
\(\mathbf{4- x = 0}\)
Collect like terms
\(\mathbf{x = 4}\)
Recall that: \(\mathbf{y= 4- \frac{x}{2}}\)
Substitute 4 for x
\(\mathbf{y = 4- \frac{4}{2}}\)
\(\mathbf{y = 4- 2}\)
\(\mathbf{y = 2}\)
So, we have:
\(\mathbf{x = 4}\) and \(\mathbf{y = 2}\)
Hence, the dimension of the rectangle that will produce the maximum area is 4 units by 2 units
Read more about maximum areas at:
https://brainly.com/question/11906003
