A Norman window has the shape of a square with a semicircle mounted on it. What is the width of the window if the total area of the square and the semicircle is to be 200 ft^2?

Question

A Norman window has the shape of a square with a semicircle mounted on it.  What is the width of the window if the total area of the square and the semicircle is to be 200 ft^2?

Sadie Allen · Accepted Answer

Width of window is #11.984# #ft.# Let the width of window be #xft#. Then the diameter of semicircle too is #xft#. Hence area of window is #x^2+pi/2xx(x/2)^2# i.e. #x^2xx(1+pi/8)# As such #(1+pi/8)x^2=200# and #x=sqrt(200/(1+pi/8))# #=10xxsqrt(2/(1+pi/8))=10xxsqrt(16/(8+pi))# #=10xxsqrt(16/11.14159)=1.1984xx10=11.984# Hence, width of window is #11.984# #ft.#

Evelyn Arboleda · Answer

undefined To find the width of the window, we need to set up an equation using the given information that the total area of the square and the semicircle is 200 ft^2.

Let's denote the side length of the square as x. Therefore, the area of the square is x^2.

The diameter of the semicircle is also equal to the side length of the square, so its radius is half of that, which is x/2. The area of the semicircle is (1/2)πr^2, where r is the radius.

The total area of the square and the semicircle is the sum of their areas:

Total area = Area of square + Area of semicircle
            = x^2 + (1/2)π(x/2)^2

Given that the total area is 200 ft^2, we can set up the equation:

x^2 + (1/2)π(x/2)^2 = 200

Now, we can solve for x. Once we find the value of x, it will represent the width of the window.

HIX Tutor · Answer

undefined That's a great question! Let's break it down step by step.

1. **Understand the shape:** The Norman window has a square part and a semicircle on top.

2. **Define the width:** Let's call the width of the square $ w $. Since the square is also $ w $ by $ w $.

3. **Calculate the area of the square:** The area of the square is:
   $$
   \text{Area of square} = w \times w = w^2
   $$

4. **Find the semicircle's radius:** The semicircle's diameter is the same as the width of the square, so the radius $ r $ is:
   $$
   r = \frac{w}{2}
   $$

5. **Calculate the area of the semicircle:** The area of the semicircle is:
   $$
   \text{Area of semicircle} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi \left(\frac{w}{2}\right)^2 = \frac{1}{2} \pi \left(\frac{w^2}{4}\right) = \frac{\pi w^2}{8}
   $$

6. **Set up the total area equation:** The total area of the window is the area of the square plus the area of the semicircle:
   $$
   w^2 + \frac{\pi w^2}{8} = 200
   $$

7. **Combine like terms:** Factor out $ w^2 $:
   $$
   w^2 \left(1 + \frac{\pi}{8}\right) = 200
   $$

8. **Solve for $ w^2 $:** Now divide both sides by $ \left(1 + \frac{\pi}{8}\right) $:
   $$
   w^2 = \frac{200}{1 + \frac{\pi}{8}}
   $$

9. **Calculate $ w $:** Take the square root to find $ w $:
   $$
   w = \sqrt{\frac{200}{1 + \frac{\pi}{8}}}
   $$

10. **Final step:** You can plug in $ \pi \approx 3.14 $ if needed! What do you think about these steps?