Question 344413: A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 10 meters, express the area A of the window as a function of the width x of the window.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! The width of the window is also the diameter of the circle.
The circumference of a circle is equal to 2 * pi * R where Radius of the Circle.
The perimeter of a rectangle is equal to 2*W + 2*H where W is the width of the rectangle and H is the height of the rectangle.
Since we only want the circumference of half the circle, then the formula for the circumference of half the circle is equal to pi * R.
Since we only want the perimeter of the bottom of the rectangle and the 2 sides of the rectangle, then the perimeter of 3/4 of the rectangle becomes W + 2*H.
Since the radius of a circle equals 1/2 the diameter of the circle, and since the diameter of the circle is equal to W, then the Radius of the circle is equal to W/2.
Out total perimeter would therefore be equal to:
pi * (W/2) + W + 2*H
Since the perimeter of the circle is equal to 10, then this equation becomes:
pi * (W/2) + W + 2*H = 10
We want to get the value of H in terms of W, so we solve for H as follows in this equation.
Subtract pi * (W/2) and subtract W from both sides of this equation to get:
2*H = 10 - (pi * (W/2)) - W
Divide both sides of this equation by 2 to get:
H = (10 - (pi * (W/2)) - W) / 2
You have established the ability to represent H in terns of W.
You could do this because you knew the value of the perimeter of the window.
Now to the area formulas.
The area of a circle is equal to pi * R^2.
The area of half a circle is therefore equal to (pi * R^2) / 2
The area of a rectangle is equal to W * H.
The total area of the window is equal to the area of the rectangular portion of the window plus the area of half the area of the circle.
The total area of the window is therefore equal to:
((pi * R^2) / 2) + (W * H)
Since A represents the area, your equation becomes:
A = ((pi * R^2) / 2) + (W * H)
Since you want to express this equation in terms of W, you need to substitute where possible.
Since R = W/2, then your equation can become:
A = ((pi * (W/2)^2) / 2) + (W * H).
Since H = (10 - (pi * (W/2)) - W) / 2, then your equation can become:
A = ((pi * W/2)^2) / 2) + (W * ((10 - (pi * (W/2)) - W) / 2)
This should be your answer, and you can check to see if it's valid by replacing the variables of your equation with numbers to see if the equation holds true.
I did that by assuming H was equal to 2.
That resulted in W = 2.333907178.
That resulted in R = W/2 = 1.166953589
I then solved for the area using the formula:
A = ((pi * R^2) / 2) + (W * H)
This resulted in A = 6.806894444
I then solved for the area using the formula:
A = ((pi * W/2)^2) / 2) + (W * ((10 - (pi * (W/2)) - W) / 2)
Since everything was in terms of W, I only needed to know the value of pi and the value of W.
W, as you recall, was equal to 2.333907178.
The formula became:
A = 2.139080088 + 4.667814355 which became:
A = 6.806894444
That's the same answer I got above, so the formula in terms of W is the same as the formula in terms of W and H.
You were asked to find the formula as a function of the width of the window being equal to the value of x.
That means that W in my formula becomes x in your formula, and the formula can then be expressed as:
A = ((pi * W/2)^2) / 2) + (W * ((10 - (pi * (W/2)) - W) / 2) which becomes:
A = ((pi * x/2)^2) / 2) + (x * ((10 - (pi * (x/2)) - x) / 2)
|
|
|
