Math Problem Statement

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?

Solution

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 99 feet.
  • The height of the rectangular part of the window is 55 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 55 feet. Thus, the radius rr of the semicircles is 52=2.5\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). 2×9+1×5=18+5=23feet.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 2πr2 \pi r, but for two semicircles, the total perimeter is equivalent to one full circle. Circumference of semicircles=2πr=2π(2.5)=5πfeet.\text{Circumference of semicircles} = 2 \pi r = 2 \pi (2.5) = 5 \pi \, \text{feet}.

Thus, the total perimeter is: 23+5π23+15.71=38.71feet.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: 9×5=45square feet.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 πr2\pi r^2: Area of semicircles=πr2=π(2.5)2=π×6.25=6.25πsquare feet.\text{Area of semicircles} = \pi r^2 = \pi (2.5)^2 = \pi \times 6.25 = 6.25 \pi \, \text{square feet}. Area of semicircles6.25×3.1416=19.63square feet.\text{Area of semicircles} \approx 6.25 \times 3.1416 = 19.63 \, \text{square feet}.

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

Final Answer:

  • Metal trim (perimeter): 38.71feet\approx 38.71 \, \text{feet}
  • Glass needed (area): 64.63square feet\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 π\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.

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Math Problem Analysis

Mathematical Concepts

Geometry
Perimeter
Area
Circles

Formulas

Perimeter of a rectangle = 2L + 2W
Circumference of a circle = 2πr
Area of a rectangle = L × W
Area of a circle = πr^2

Theorems

Circle Circumference Theorem
Area of Circle Theorem

Suitable Grade Level

Grades 6-8