A picture window has a length of 

9 

feet and a height of 

5 

​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 problem, we need to calculate two things:

1. **The perimeter of the window**: This includes the length and height of the rectangular part as well as the perimeter of the semicircular caps on both ends.
2. **The area of the window**: This includes the area of the rectangular part and the area of the semicircles on both ends.

### Given:
- The **length** of the rectangular part of the window is \( 9 \) feet.
- The **height** of the rectangular part of the window is \( 5 \) feet.
- There are **two semicircular caps** on the window ends. The **diameter** of each semicircular cap is equal to the height of the rectangle, which is \( 5 \) feet. Thus, the radius \( r \) of the semicircles is \( \frac{5}{2} = 2.5 \) feet.

### Step 1: **Perimeter Calculation (Metal Trim Needed)**

The perimeter consists of:
- The perimeter of the rectangular part: Two lengths and one height (the top and bottom heights are accounted for by the semicircular caps).
  \[
  2 \times 9 + 1 \times 5 = 18 + 5 = 23 \, \text{feet}.
  \]
- The circumference of two semicircles: The formula for the circumference of a full circle is \( 2 \pi r \), but for two semicircles, the total perimeter is equivalent to one full circle.
  \[
  \text{Circumference of semicircles} = 2 \pi r = 2 \pi (2.5) = 5 \pi \, \text{feet}.
  \]

Thus, the total perimeter is:
\[
23 + 5 \pi \approx 23 + 15.71 = 38.71 \, \text{feet}.
\]

### Step 2: **Area Calculation (Glass Needed)**

The area consists of:
- The area of the rectangular part: 
  \[
  9 \times 5 = 45 \, \text{square feet}.
  \]
- The area of the two semicircles, which is equivalent to the area of a full circle. The formula for the area of a circle is \( \pi r^2 \):
  \[
  \text{Area of semicircles} = \pi r^2 = \pi (2.5)^2 = \pi \times 6.25 = 6.25 \pi \, \text{square feet}.
  \]
  \[
  \text{Area of semicircles} \approx 6.25 \times 3.1416 = 19.63 \, \text{square feet}.
  \]

Thus, the total area is:
\[
45 + 19.63 = 64.63 \, \text{square feet}.
\]

### Final Answer:
- **Metal trim (perimeter)**: \( \approx 38.71 \, \text{feet} \)
- **Glass needed (area)**: \( \approx 64.63 \, \text{square feet} \)

Would you like more details or have any questions?  

Here are some related questions you might find helpful:
1. How would the trim change if the height of the window were doubled?
2. What if the window had no semicircular caps? How would the perimeter and area change?
3. How does the value of \( \pi \) affect the perimeter and area in this problem?
4. What is the perimeter of the rectangular part alone?
5. How do we calculate the area and perimeter for different shapes like ellipses or triangles?

**Tip**: Always make sure to distinguish between the perimeter and area when dealing with shapes—perimeter measures the boundary, while area measures the surface inside.

A picture window has a length of 9 feet and a height of 5 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 a rectangular base of 9 feet by 5 feet and semicircular caps. This detailed solution includes key geometric concepts such as the perimeter and area formulas for rectangles and circles, suitable for Grades 6-8.

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