Anorman window is constructed by adjoining a semicircle to the top of an ordinary rectangular. find the dimensions of a norman window of maximum area if the total perimeter is 38 feet?

The value of x so that the greatest possible amount of light is admitted is 5.32 ft.

Let x represent the radius of the semicircle, hence the width of the rectangle = 2x. Also, the length of the rectangle is y

The perimeter of the window = πx + 2y + 2x = x(π + 2) + 2y

38 = x(π + 2) + 2y

The area of the window (A) = 0.5πx² + 2yx

Hence the value of x so that the greatest possible amount of light is admitted is 5.32 ft.

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Step-by-step explanation:

Please find the attachment.

We have been given that a norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular. The total perimeter is 38 feet.

The perimeter of the window will be equal to three sides of rectangle plus half the perimeter of circle. We can represent our given information in an equation as:

We can see that diameter of semicircle is W. We know that diameter is twice the radius, so we will get:

Let us find area of window equation as:

Now, we will solve for L is terms W from perimeter equation as:

Substitute this value in area equation:

Since we need the area of window to maximize, so we need to optimize area equation.

Let us find derivative of area equation as:

To find maxima, we will equate first derivative equal to 0 as:

Upon substituting in equation , we will get:

Therefore, the dimensions of the window that will maximize the area would be and .

the graph in the attached figure

step-by-step explanation:

we have

using a graphing tool

the graph in the attached figure

the roots are ( x-intercepts) x=-0.5,x=1,x=3

the y-intercept is the point (0,3)