A window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the
window is 30ft, express the area of the window as a function of the width of the window.
let the width of the window be x
Let the length of the window be y, which makes
the radius of the semicircle y/2
perimeter = 2x + y + πy = 30
y(1 + π) = 30-2x
y = (30 - 2x)/(1 + π)
area = xy + (1/2)π r^2
= x(30-2x)(1+π) + (1/2)π([(30-2x)/(1+π)]^2 / 4
Suppose we define the width as x and the length as 2y, which makes
the radius of the half-circle as y
then we get
2x + 2y + 2πy = 30
x + y + πy = 15
y(1+π) = 15-x
y = (15-x)/(1+π)
area = 2xy + (1/2)π y^2
= 2x(15-x)/(1+π) + (1/2)π [(15-x)/(1+π)]^2
take your pick, but check my algebra for each
To express the area of the window as a function of the width, let's first break down the shape of the window and find its dimensions.
The window consists of a rectangle surmounted by a semicircle. Let's say the width of the rectangle is "w" feet. Since the rectangle is surmounted by a semicircle, the diameter of the semicircle is also equal to the width "w" of the rectangle.
Now, let's find the length of the rectangle. Since the given perimeter of the window is 30ft and the perimeter of a rectangle is given by the formula P = 2(length + width), we can set up the equation:
30 = 2(length + w)
Simplifying the equation, we have:
15 = length + w
But we have one more piece of information - the length of the rectangle is equal to the diameter of the semicircle, which is equal to the width "w". So we can substitute the value of length with the value of "w" in the equation:
15 = w + w
15 = 2w
Solving for "w", we find:
w = 7.5
Now that we know the dimensions of the width and length of the rectangle, we can calculate the area of the complete window. The area of the rectangle is given by the formula A = length * width. In this case, the length and width of the rectangle are both "w", so we can substitute the values into the equation:
A = w * w
A = w^2
Therefore, the area of the window as a function of the width "w" is given by A = w^2.