A picture window has a length of 10 feet in a height of 9 feet with a semicircle cap on each end how much metal trim is needed for the perimeter of the entire window and how much glasses needed for the opening of the window

We need to calculate two things for this window:

1. **Metal Trim for the Perimeter:** This includes both the straight edges and the semicircles on each end.
2. **Glass Area for the Opening:** This includes the rectangular part and the areas of the two semicircles.

### Step 1: Metal Trim (Perimeter)
The window consists of a rectangle and two semicircles.

- **Perimeter of the Rectangle:**  
  The rectangle has a length of 10 feet and a height of 9 feet.  
  Perimeter of the rectangle (not including the semicircles):  
  \[
  P_{\text{rect}} = 2 \times (10 + 9) = 38 \text{ feet}
  \]

- **Perimeter of the Semicircles:**  
  The two semicircles together form a full circle.  
  The diameter of each semicircle is equal to the width of the rectangle, which is 10 feet.  
  The circumference of a full circle is \(C = \pi \times d\), where \(d\) is the diameter.  
  So the perimeter of the two semicircles is:  
  \[
  P_{\text{semi}} = \pi \times 10 \approx 31.42 \text{ feet}
  \]

- **Total Metal Trim (Perimeter):**  
  Adding the perimeter of the rectangle and the semicircles:  
  \[
  P_{\text{total}} = P_{\text{rect}} + P_{\text{semi}} = 38 + 31.42 \approx 69.42 \text{ feet}
  \]

### Step 2: Glass Area
We need to find the area of the rectangular part and the two semicircles.

- **Area of the Rectangle:**  
  The area of the rectangle is:  
  \[
  A_{\text{rect}} = 10 \times 9 = 90 \text{ square feet}
  \]

- **Area of the Semicircles:**  
  The two semicircles together form a full circle.  
  The area of a full circle is \(A = \pi \times r^2\), where \(r\) is the radius.  
  The radius of each semicircle is \(r = \frac{10}{2} = 5\) feet.  
  So the area of the two semicircles is:  
  \[
  A_{\text{semi}} = \pi \times 5^2 = \pi \times 25 \approx 78.54 \text{ square feet}
  \]

- **Total Glass Area:**  
  Adding the area of the rectangle and the semicircles:  
  \[
  A_{\text{total}} = A_{\text{rect}} + A_{\text{semi}} = 90 + 78.54 \approx 168.54 \text{ square feet}
  \]

### Final Answers:
- **Metal Trim Needed:** \(69.42\) feet
- **Glass Area Needed:** \(168.54\) square feet

Would you like further explanation or more details?

Here are 5 related questions to expand this topic:
1. How would the perimeter change if the window had a different diameter for the semicircles?
2. What is the perimeter and area if only one semicircle caps the window?
3. How would you calculate the metal trim if the window had elliptical caps instead of semicircles?
4. What is the perimeter and area if the window is scaled up by 50% in both dimensions?
5. How would the metal trim calculation differ if the window had a different shape, such as an arch?

**Tip:** When dealing with shapes combining rectangles and circles, always treat the curved and straight parts separately for accurate calculations.

A picture window has a length of 10 feet and a height of 9 feet with a semicircle 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?

This solution covers the perimeter and glass area calculation of a picture window measuring 10 feet in length and 9 feet in height, with semicircular caps at each end. It includes key geometric concepts such as the perimeter of rectangles and circles and is suitable for students in Grades 8-10.

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