A picture window has a length of 12 feet and a height of 10 ​feet, with a semicircular cap on each end​ (see the​ figure). How much metal trim is needed for the perimeter of the entire​ window, and how much glass is needed for the opening of the​ window?

To solve this, we'll break the problem down into two parts: calculating the perimeter for the metal trim and calculating the area for the glass.

### 1. **Metal Trim (Perimeter Calculation)**

The window consists of a rectangular part and two semicircles on each end.

- **Length of the rectangular part**: 12 feet
- **Height of the rectangular part**: 10 feet

The perimeter of the rectangular part includes the height on both sides and the base. However, the top and bottom of the window are semicircles, so the entire length is already accounted for by the semicircles.

- **Semicircles**: Since there are two semicircles, their combined perimeter equals the circumference of a full circle. The diameter of the semicircles equals the width of the window, which is 12 feet.

The circumference of a full circle is \(C = \pi d\), where \(d\) is the diameter. For this case:

\[
C_{\text{circle}} = \pi \times 12
\]

Thus, the total perimeter is:

\[
P = 2 \times 10 + \pi \times 12
\]

### 2. **Glass Area (Opening of the Window)**

The area of the window consists of the area of the rectangular part and the area of the two semicircles (which together form a full circle).

- **Area of the rectangular part**: 
\[
A_{\text{rectangle}} = \text{length} \times \text{height} = 12 \times 10
\]
- **Area of the full circle**: The area of a circle is \(A = \pi r^2\), and the radius \(r\) is half the diameter of 12 feet, so \(r = 6\).

\[
A_{\text{circle}} = \pi \times 6^2
\]

Thus, the total area is:

\[
A_{\text{total}} = A_{\text{rectangle}} + A_{\text{circle}}
\]

Now, let's calculate the values for the perimeter and the area.### Results:

1. **Metal Trim (Perimeter)**: The total perimeter needed for the metal trim around the window is approximately \( 57.70 \) feet.
   
2. **Glass (Area)**: The total area of the glass required for the window is approximately \( 233.10 \) square feet.

Would you like further details or have any other questions?

Here are 5 related questions:
1. How would the perimeter change if the window were wider by 2 feet?
2. What is the impact on glass area if the height of the rectangular part increases by 3 feet?
3. How would the perimeter formula differ if the window had a quarter circle cap instead of a semicircle?
4. How can you find the perimeter if the top and bottom caps were elliptical instead of circular?
5. How would you calculate the amount of metal trim if only one semicircular cap was present?

**Tip**: For quick approximations involving circles, use \( \pi \approx 3.14 \) when exact values aren't necessary.

A picture window has a length of 12 feet and a height of 10 ​feet, with a semicircular cap on each end​. How much metal trim is needed for the perimeter of the entire​ window, and how much glass is needed for the opening of the​ window?

Learn how to calculate the perimeter and area of a picture window with semicircular caps. This problem involves geometry concepts like the perimeter of a rectangle, the circumference of circles, and area calculations. Suitable for students in Grades 9-12.

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