A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 26 ft, express the area A of the window as a function of the width x of the window.
To find the area A of the window as a function of the width x, we need to break down the window into its rectangular and semicircular parts.
Let's start by determining the dimensions of the rectangle. The width of the rectangle is x, which means that the length of the rectangle will be 2x (twice the width).
The perimeter of the rectangle is given by the formula: Perimeter = 2(length + width)
Since we know the perimeter is 26 ft, we can write the equation as:
26 = 2(2x + x)
Simplifying the equation:
26 = 2(3x)
26 = 6x
Dividing both sides by 6:
x = 4.33 ft
Now, let's find the area of the rectangular part:
Area of rectangle = length * width
Area of rectangle = (2x) * x
Area of rectangle = 2x^2
Next, we calculate the area of the semicircle. The diameter of the semicircle will be equal to the width of the rectangle since the semicircle is surmounted on the rectangle. The radius of the semicircle will be half the diameter or x/2.
The formula for the area of a semicircle is: Area of semicircle = 1/2 * π * r^2
Area of semicircle = 1/2 * π * (x/2)^2
Area of semicircle = 1/2 * π * (x^2)/4
Area of semicircle = π * x^2 / 8
Finally, we can calculate the total area A by adding the area of the rectangle and the area of the semicircle:
A = 2x^2 + π * x^2 / 8
A = (16x^2 + πx^2) / 8
A = (16 + π/8) * x^2 / 8
Thus, the area A of the Norman window as a function of the width x is:
A(x) = (16 + π/8) * x^2 / 8
To express the area of the window as a function of the width, let's break down the window shape into its components, namely the rectangle and the semicircle.
Let's start by defining the width of the rectangle as 'x'. Since the rectangle also acts as the base of the semicircle, the length of the rectangle will be 'x' as well.
The perimeter of the window is given as 26ft. We can express it as the sum of the perimeter of the rectangle and half the circumference of the semicircle.
Perimeter of the rectangle = 2(width + length) = 2(x + x) = 4x ft
Circumference of the semicircle = π * radius = π * (width / 2) = π * (x/2)
So, the equation for the perimeter is:
4x + (π * (x/2)) = 26
Next, we solve the equation for 'x' to find the width of the window.
4x + (π * (x/2)) = 26
To solve for 'x', we first multiply the entire equation by 2 to eliminate the fraction:
8x + πx = 52
Next, we combine the like terms:
(8 + π)x = 52
To isolate 'x', we divide both sides of the equation by (8 + π):
x = 52 / (8 + π)
Now, we have an expression for the width 'x'.
To express the area of the window (A) in terms of 'x', we need to calculate the sum of the area of the rectangle and the area of the semicircle.
Area of the rectangle = length * width = x * x = x^2
Area of the semicircle = (1/2) * π * (width/2)^2 = (1/2) * π * (x/2)^2 = (1/2) * π * (x^2/4) = (π/8) * x^2
Therefore, the area A of the window is given by the sum of the area of the rectangle and the area of the semicircle:
A(x) = x^2 + (π/8) * x^2
Simplifying the expression further:
A(x) = (1 + π/8) * x^2
So, the area A of the window is expressed as a function of the width x.
Now I would have started by defining the radius as r and the height of the rectangular window as h ( avoiding a lot of fractions)
Perimeter of half a circle = (1/2)(2πr)
= πr
the three sides of the rectangle = 2r + 2h
so
2r + 2h + πr = 26
h = (26 - 2r - πr)/2 = 13 - r - πr/2
area = (1/2)πr^2 + 2rh
= (1/2)πr^2 + 2r(13-r-πr/2)
= (1/2)πr^2 + 26r - 2r^2 - πr^2
= 26r - 2r^2 - (πr^2)/2
now if we let the base of the window be x
then the radius is x/2
let the height of the rectangle be y
(1/2)(2π(x/2)) + x + 2y = 26
(πx)/2 + x + 2y = 26
y = (26 - x - πx/2)/2
area = (1/2)π(x/2)^2 + xy
= πx^2/8 + x(26 - x - πx/2)/2
= πx^2/8 + 13x - x^2/2 - πx^2/4
= - πx^2/8 + 13x - x^2/4
check each setup for algebraic errors, I did not write it out first.