The base length that will maximize the area for such a window is 168.03 cm. The exact largest value of x when this occurs is 233.39 cm
Suppose we make an assumption that:
- (x) should be the width of the rectangle base;
- (h) should be the height of the rectangle
Also, provided that the diameter of the semi-circle appears to be the base of the rectangle, then;
- the radius
and, the perimeter of the window can now be expressed as:
Given that the perimeter = 600 cm
∴
Since h > 0, then:
By rearrangement and using the inverse rule:
Thus, the largest length x = 233.39 cm
However, the area of the window is given as:
Now, at maximum, when the area A = 0. Taking the differentiation, we have:
Making x the subject of the formula, we have:
x = 168.03 cm
Taking the second derivative:
Therefore, we can conclude that the maximum area that exists for such a window is 168.03 cm
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