62. A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of...

Summary

This educational video explains how to express the area of a Norman window, which is a rectangle topped by a semicircle, as a function of its width. With a given perimeter of 30 units, the video demonstrates the process of finding the dimensions of the window and then calculating the area. It involves solving for the length of the rectangle and the radius of the semicircle, and then using these to derive the area formula. The final expression for the area in terms of the width 'x' is given as \(15x - \frac{x^2}{2} + \frac{\pi}{8}x^2\), which combines the area of the rectangular part and the circular segment.

Takeaways

• 📐 The problem involves a Norman window with a rectangular base and a semicircular top.
• 🔢 The task is to express the area of the window as a function of its width, denoted by x.
• ➕ The perimeter of the window is given as 30 units.
• 🧮 The perimeter equation includes the width x, twice the length L, and half the circumference of the semicircle.
• 🔄 The radius of the semicircle is half of the width x.
• ✂️ The perimeter equation simplifies to solve for L in terms of x.
• 🔄 Substituting L into the area formula gives the area in terms of x.
• 🟦 The area is calculated by adding the area of the rectangle (L * x) to half the area of the circle.
• 🔗 After simplifying, the final area equation is in terms of x and includes both linear and quadratic terms.
• 📊 The area formula can be used to analyze the shape and size of the window based on different values of x.

Q & A

What is the shape of the window described in the video?

The window is in the shape of a rectangle surmounted by a semicircle.

What is the perimeter of the window?

The perimeter of the window is 30 units.

How is the perimeter of the semicircle related to the width of the window?

The perimeter of the semicircle is half of the circumference of a full circle with a diameter equal to the width of the window (x), which is \(\pi x / 2\).

What is the relationship between the width of the window (x) and the radius of the semicircle?

The radius of the semicircle is half of the width of the window, so the radius is \(x / 2\).

How do you express the length (L) of the rectangular part of the window in terms of the width (x)?

The length (L) can be expressed as \(L = (30 - x - \pi x / 2) / 2\) after simplifying the perimeter equation.

What is the formula for the area of the window in terms of the width (x)?

The area of the window in terms of the width (x) is \(A = 15x - \pi x^2 / 8\).

Why is the area of the circle part of the window not squared when calculating the area?

The area of the circle is calculated using the formula \(\pi r^2\), where r is the radius. The radius is \(x / 2\), so the area is \(\pi (x / 2)^2\), which simplifies to \(\pi x^2 / 4\), not squared by the width x again.

How does the area of the rectangle part of the window contribute to the total area?

The area of the rectangle part is calculated by multiplying the length (L) by the width (x), which is \(L \times x\).

What is the significance of the term \(\pi x^2 / 8\) in the area formula?

The term \(\pi x^2 / 8\) represents the area of the semicircle part of the window, accounting for half of the circle's area with radius \(x / 2\).

How is the total area of the window calculated?

The total area of the window is calculated by adding the area of the rectangular part and the area of the semicircle part, which is \(15x - \pi x^2 / 8\).