A Norman window has the shape of a rectangle surmounted by a semicircle as in the figure below. If the perimeter of the window is 37 ft, express the area, A, as a function of the width, x, of the window.
How do you do this problem?
Perimeter=w+2h+pi*w.2
area=lw+1/2 PI (w/2)^2
now go to the perimeter equation, and solve for l, then put it in the area equation?
60.1
To solve this problem, we need to break it down into smaller steps. Here's how you can approach it:
1. Understand the problem: We are given a Norman window, which consists of a rectangle with a semicircle on top. The perimeter of the window is 37 ft, and we need to express the area, A, as a function of the width, x, of the window.
2. Visualize the Norman window: Draw a diagram of the window to get a clear picture of its shape. Label the width of the rectangular part as x and the height as y.
3. Find the perimeter: The perimeter of the window is the sum of the lengths of all the sides. In this case, it consists of the four sides of the rectangle and half the circumference of the semicircle. So, we have:
Perimeter = 2x + y + πr/2, where r is the radius of the semicircle and is equal to x/2.
4. Substitute the given perimeter: We are given that the perimeter is 37 ft, so we can write the equation as:
37 = 2x + y + π(x/2)/2.
5. Simplify the equation: Distribute π/2 into (x/2) to get (πx/4). Rearrange the equation to isolate y by subtracting 2x and (πx/4) from both sides:
y = 37 - 2x - (πx/4).
6. Calculate the area: The area of the window is the sum of the area of the rectangle and half the area of the semicircle. The area of the rectangle is given by length multiplied by width, which in this case, is x * y. The area of the semicircle is (πr^2)/2, where r is again x/2. So we have:
A = xy + (π/2)(x/2)^2.
7. Substitute the expression for y: We can substitute the expression we found for y in step 5 into the equation for the area:
A = x(37 - 2x - (πx/4)) + (π/2)(x/2)^2.
8. Simplify the equation: Distribute x into (37 - 2x - (πx/4)) to get 37x - 2x^2 - (πx^2/4) and simplify further if needed.
By following these steps, you can express the area, A, of the Norman window as a function of its width, x.