We need to find the height $h$ of a stained-glass window that is shaped like a rectangle surmounted by a semicircle. The width of the window is given as 3 feet, and the total area of the glass is 26 square feet.

Step-by-Step Process:

1. Define the components of the window:

   • The width of the window is given as $3$ feet.
   • The height $h$ represents the vertical height of the rectangular portion of the window.
   • The semicircle sits on top of the rectangle, and its diameter equals the width of the window, i.e., $3$ feet.

2. Area of the rectangular portion: The area $A_{\text{rect}}$ of the rectangular part of the window is given by: $A_{\text{rect}} = \text{width} \times \text{height} = 3 \times h = 3h$

3. Area of the semicircular portion: The diameter of the semicircle is 3 feet, so the radius $r$ of the semicircle is: $r = \frac{3}{2} = 1.5 \text{ feet}$ The area $A_{\text{semi}}$ of the semicircle is half the area of a full circle, which is given by: $A_{\text{semi}} = \frac{1}{2} \times \pi r^2 = \frac{1}{2} \times \pi (1.5)^2 = \frac{1}{2} \times \pi \times 2.25 = 1.125\pi$

4. Total area of the window: The total area $A_{\text{total}}$ is the sum of the areas of the rectangle and the semicircle: $A_{\text{total}} = A_{\text{rect}} + A_{\text{semi}} = 3h + 1.125\pi$

5. Set up the equation: The total area of the window is given as 26 square feet. Therefore, we can set up the following equation: $3h + 1.125\pi = 26$ Solving for $h$, we first subtract the area of the semicircle from both sides: $3h = 26 - 1.125\pi$

6. Approximate the value of $\pi$: Using $\pi \approx 3.1416$, we can calculate the right-hand side: $3h = 26 - 1.125 \times 3.1416 = 26 - 3.534 = 22.466$

7. Solve for $h$: Finally, divide both sides by 3 to find $h$: $h = \frac{22.466}{3} \approx 7.489$

Thus, the height of the rectangular portion of the window is approximately $h \approx 7.49$ feet.

Would you like further details or have any questions?

Related Questions:

1. How would the height change if the total area of the glass were increased to 30 square feet?
2. What would be the height if the width of the window were 4 feet instead of 3 feet?
3. How can we find the radius of the semicircle given the total area and width?
4. If the window had a different shape, like an ellipse on top, how would the calculations change?
5. What is the exact area of the semicircular part, without approximating $\pi$?

Tip:

When solving geometry problems, breaking the figure into simpler shapes and calculating their areas individually helps streamline the solution process.