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 \(15x - \pi x^2 / 8\).

Outlines

00:00

🔍 Calculating Window Area from Perimeter

This paragraph introduces a mathematical problem involving a Norman window, which is a rectangle topped by a semicircle. The challenge is to express the area of the window as a function of its width 'x', given the total perimeter is 30 units. The solution begins by breaking down the perimeter into the width 'x', the length 'L' of the rectangle, and the semicircle's contribution. The radius of the semicircle is determined to be x/2, and its perimeter contributes half of πx to the total perimeter. The equation for the perimeter is set up as 2L + x + (π * x) / 2 = 30. Solving for 'L' gives L = (30 - x - (π * x) / 2) / 2. The area of the window is then calculated by summing the area of the rectangle (L * x) and the area of the semicircle (π * (x/2)^2). After substituting the expression for 'L' and simplifying, the area 'A' in terms of 'x' is found to be A = 15x - (π * x^2) / 8.

Keywords

💡Perimeter

Perimeter refers to the total length around a two-dimensional shape. In the video, the perimeter of the window is given as 30 units, which includes both the rectangular and the semicircular parts. The script uses the perimeter to establish an equation involving the width 'x' and the length 'L' of the rectangle, which is crucial for calculating the area of the window.

💡Rectangle

A rectangle is a quadrilateral with four right angles. The video describes the window's shape as a rectangle surmounted by a semicircle. The rectangle's width 'x' and length 'L' are used to calculate the perimeter and subsequently the area, making the rectangle a fundamental part of the problem-solving process.

💡Semicircle

A semicircle is half of a circle. The video script describes the window's shape as having a semicircle on top of a rectangle. The semicircle contributes to the perimeter and the area of the window, with the script specifically calculating half of the circle's perimeter as part of the total window perimeter.

💡Area

Area is the amount of space inside the boundary of a two-dimensional shape. The main objective of the video is to express the area 'A' of the window as a function of its width 'x'. The area is calculated by summing the area of the rectangle and the area of the semicircle, which involves integrating the geometrical properties of both shapes.

💡Width

Width is the measurement across the shorter dimension of a rectangle. In the video, 'x' represents the width of the window's rectangular part. The width is a key variable in the equations derived to find the perimeter and area of the window, as it directly influences both measurements.

💡Length

Length is the measurement along the longest dimension of a rectangle. In the video, 'L' is used to denote the length of the rectangle, which is found in relation to the width 'x'. The length is essential for calculating the perimeter and area of the window, as it is one of the dimensions of the rectangle.

💡Radius

Radius is the distance from the center of a circle to any point on its edge. The script establishes that the radius of the semicircle is half the width 'x' of the rectangle. This relationship is used to calculate the perimeter and area contributions of the semicircle to the total window measurements.

💡Circumference

Circumference is the distance around a circle. The video script calculates half the circumference of the semicircle to determine its contribution to the window's perimeter. The formula used is π times the diameter, which is then halved because only a semicircle is present.

💡Pi (π)

Pi, often denoted as π, is a mathematical constant representing the ratio of a circle's circumference to its diameter. In the video, π is used in the formulas for calculating the circumference of the semicircle and the area of the circle, reflecting its importance in circle-related geometry.

💡Function

A function in mathematics is a relation between a set of inputs and a set of possible outputs. The video aims to express the area 'A' of the window as a function of the width 'x'. This means finding a formula where 'A' is calculated based on the value of 'x', which is a fundamental concept in algebra and calculus.

💡Equation

An equation is a statement that two expressions are equal. The video script involves setting up and solving equations to relate the perimeter, area, width, and length of the window. Equations are used to express the relationships between these variables and to solve for unknowns, such as the length 'L' in terms of width 'x'.

Highlights

The problem involves calculating the area of a Norman window with a rectangular shape and a semicircle on top.

The perimeter of the window is given as 30 units.

The perimeter equation includes the width (x), the length (L), and half the circumference of the semicircle.

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

The perimeter of the semicircle is π times the radius, but only half of it contributes to the total perimeter.

The total perimeter is expressed as 30 = 2L + x + (π * x) / 2.

Solving for L gives L = (30 - x - (π * x) / 2) / 2.

The area of the window is the sum of the area of the rectangle and the area of the semicircle.

The area of the rectangle is L * x, where L is expressed in terms of x.

The area of the semicircle is (π * (x/2)^2) / 2.

The final area equation is A = 15x - (π * x^2) / 8.

The area is expressed solely in terms of the window's width (x).

The area calculation combines linear and quadratic terms in x.

The video demonstrates a step-by-step approach to solving geometry problems involving mixed shapes.

The presenter uses algebraic manipulation to express all terms in terms of the width (x).

The final area formula is a function of x, showcasing the relationship between the window's dimensions and its area.

Transcripts

hello and welcome back to another

video in this problem we have a Norman

window has the shape of a rectangle

surmounted by a semicircle demonstrated

by this diagram here we're asked if the

perimeter of the window is 30t Express

the area a of the window as a function

of the width x of the

window so to do this we're going to take

the perimeter and say how do we find

this side by

side so the

perimeter which is 30

this is equal to what we have this side

the width x we have this side of the

rectangular part before it becomes a

circle this side we can just call

L plus the perimeter of the

circle we have a circle here with the

center here and this is the

radius there's also a radius in this

direction so two times the radius

is equal to X therefore the radius is

equal to x

2 so

the circumference of this

circle the perimeter of it is equal to

Pi times the

diameter which is equal to twice the

radius which is equal to

X so * X however we only have half of

the circle that's exposed the other half

is within the circle or is within the

window so it's not part of the um

circumference not part of the

perimeter so only half the circle

contributes to our perimeter so

therefore we have to divide by two and

then finally we have another L the same

length down the other

side how is this going to help us get

the area well we can find the area as is

in terms of X and L but we only want it

in terms X so we have to solve for l in

this equation so 30 is equal L + L 2 L +

x + < * x /

2 then we subtract X and we subtract < *

x/ 2 from both

sides these cancel we're off with 2 * L

is equal to 30 - x - < * x /

2 dividing for

L these cancel the length L is equal to

30 / 2 is 15 x over 2 there's no way to

simplify and then pix over 2 / 2 again

is - pkx over

4 so now we have this length in terms of

the width

x so now we can use the area of equation

which we

have the area is equal to the area of

this rectangle

which is the length time the width L *

X plus 12 of the area of this

circle what is the area of the circle PK

R 2K * the radius squ so Pi * the radius

x

over2 squared make sure not to square

the

pi this can be turned into L is 15-

x / 2 - < x over 4

plus

pi

* 12 * 12 2 which is 12 * 1/4 is

1/8

x^2 and sure this should betimes

X so this is equal to 15 *

X and if it's unclear why I'm

multiplying by X because I forgot it in

writing it down this is L so this is L *

X is this thing right here so now we're

Distributing 15 * x - 15 or sorry - x *

X over 2 is x^2 /

2us i x * X is

x^2 over

4 plus < * 1/ 18 is < / 8 x^2

and one little thing at the end that we

can do here is combine these two so 15x

- x^2 over two stays the same we have

1/4

pi+ 1/8

Pi with an X2

attached becomes 1/8 Pi so - PK / 8

x^2 and this is the area of this

particular window in terms of X the

width