Welcome to the article A stained glass window is being designed in the shape of a rectangle surmounted by a semicircle as shown in the figure The width of. On this page, you will learn the essential and logical steps to better understand the topic being discussed. We hope the information provided helps you gain valuable insights and is easy to follow. Let’s begin the discussion!
Answer :
The height of the rectangular part of the stained glass window is approximately 4.48857 feet.
To solve for the height (h) of the stained glass window, we need to determine the total area of the window, which consists of a rectangle surmounted by a semicircle. Here are the steps to find the height:
Identify Given Data:
Width of the rectangle (and diameter of the semicircle): 3 feet
Total area of the glass used: 17 square feet
Calculate the Area of the Semicircle:
The radius (r) of the semicircle is half of the width (diameter), so:
[tex]r = \frac{3}{2} = 1.5 \text{ feet}[/tex]
The formula for the area of a semicircle is:
[tex]A_{\text{semicircle}} = \frac{1}{2} \pi r^2[/tex]
Substitute the radius:
[tex]A_{\text{semicircle}} = \frac{1}{2} \pi (1.5)^2 = \frac{1}{2} \pi (2.25) = 1.125 \pi[/tex]
Express the Area of the Rectangle:
The area of the rectangle is given by:
[tex]A_{\text{rectangle}} = \text{width} \times \text{height} = 3h[/tex]
Set Up the Equation:
The total area of the window is the sum of the areas of the rectangle and the semicircle. Therefore:
[tex]3h + 1.125\pi = 17[/tex]
Solve for h:
First, isolate the term with h:
[tex]3h = 17 - 1.125\pi[/tex]
Next, solve for h by dividing both sides by 3:
[tex]h = \frac{17 - 1.125\pi}{3}[/tex]
Calculate the Numerical Value:
Substitute the value of [tex]\pi \approx 3.14159[/tex]:
[tex]h = \frac{17 - 1.125 \times 3.14159}{3} \approx \frac{17 - 3.53429}{3} \approx \frac{13.46571}{3} \approx 4.48857 \text{ feet}[/tex]
Thank you for reading the article A stained glass window is being designed in the shape of a rectangle surmounted by a semicircle as shown in the figure The width of. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
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Rewritten by : Brahmana