A norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window 10 m, express the area A of the window as the function of width x of the window.
let the width of the rectangle be x m
let the base of the rectangle be 2y m, making the radius of the semicircle equal to y m
(my semicircle sits on the y )
2x + 2y + (1/2)(2πy) = 10
2x + 2y + πy = 10
y(2 + π) = 10 - 2x
y = (10-2x)/(2+π)
area = 2x(2y) + (1/2)πy^2
= 4xy + (1/2)πy^2
= 4x(10-2x)/(2+π) + (1/2)π(100 - 40x + 4x^2)/(4 + 4π + π^2)
clean it up if need be
Note: I considered the width to be the vertical side of the rectangle. If the intent was to call the base x m, make the necessary changes in the algebra. The steps would be the same.
It this circumference of a circle is 44cm. find its area
To express the area A of the Norman window as a function of its width x, let's start by visualizing the shape.
The Norman window consists of a rectangle surmounted by a semicircle. The perimeter of the window is 10 m, which means that the sum of the lengths of all sides of the shape equals 10 m.
Let's denote the width of the rectangle as x. Since the rectangle is surmounted by a semicircle, the height of the rectangle will be the radius of the semicircle.
The perimeter of the rectangle is calculated by summing the lengths of its four sides, which are: x, x, (2 * radius), and (2 * radius).
Perimeter = x + x + (2 * radius) + (2 * radius) = 10 m
Simplifying the equation, we have:
2x + 4radius = 10
To express the radius in terms of x, let's subtract 2x from both sides:
4radius = 10 - 2x
Now, divide both sides by 4:
radius = (10 - 2x) / 4
The area A of the Norman window is calculated by finding the sum of the areas of the rectangle and the semicircle. The area of the rectangle is given by length multiplied by width, which is x * x = x^2. The area of the semicircle is given by (1/2) * pi * radius^2.
Therefore, the equation for the area A of the Norman window in terms of the width x is:
A = x^2 + (1/2) * pi * (radius)^2
Substituting the value of radius from earlier:
A = x^2 + (1/2) * pi * [(10 - 2x) / 4]^2
Simplifying further, we have:
A = x^2 + (1/2) * pi * [(10 - 2x)^2 / 16]
This equation represents the area A of the Norman window as a function of its width x.
To find the area of the Norman window as a function of width x, first, we need to understand the geometry of the window. The window consists of a rectangle surmounted by a semicircle.
Let's break down the components of the window:
1. Rectangle: The rectangle forms the bottom part of the window and has a width x. Since the perimeter is given as 10 m, we know that the sum of all four sides of the rectangle is 10 m. Also, opposite sides of the rectangle are equal, so each of the two smaller sides measures (10 - x)/2.
2. Semicircle: The semicircle forms the top part of the window, and its diameter is equal to the width of the rectangle, which is x.
Now, we can calculate the perimeter of the window based on the sum of all the sides:
Perimeter = Length of Rectangle + Width of Rectangle + Circumference of Semicircle
10 m = x + (10 - x)/2 + πx/2
Simplifying the equation:
10 = x + 10/2 - x/2 + πx/2
10 = 2x + 10 - x + (π/2)x/2
Multiplying the equation by 2 to eliminate the fraction:
20 = 4x + 20 - x + πx/2
Rearranging the terms:
0 = 4x - x + πx/2
0 = 3x + (π/2)x/2
0 = (6/2)x + (π/4)x/2
0 = (6 + (π/4)x)/2x
Now, let's solve the equation to find the value of x:
0 = 6 + (π/4)x
(π/4)x = -6
x = -24/π
Since the width of a window cannot be negative, we discard this solution and conclude that there is no valid width for the Norman window that satisfies a perimeter of 10 m.
Therefore, there is no area A as a function of width x for the Norman window with a perimeter of 10 m.