Norman Window

harpazo

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Section R.3
Geometry Essentials
Michael Sullivan
Textbook: College Algebra Edition 9

A norman window has the shape of a rectangle surmounted by a semicircle. Find the area of the Norman window if the length of the rectangle part is 6 inches and the width is 4 inches. How much wood frame is needed to enclose the window?

Solution:

Let me see how far I can go with this problem. Maybe I can slowly work my way to the answer.

Area of rectangle part of window is 6 inches x 4 inches or 24 (inches)^2.

Area of semicircle part of window is 1/2 (π r^2).

A = 1/2 • π (2 inches)^2

A = 1/2(4π) inches

A = 2π (inches)^2

Area of Norman window is
24 (inches)^2 + 2π (inches)^2.

Is this correct?

How much wood frame is needed to enclose the window?

Let P = perimeter of rectangle part of window.

P = 2l + 2w

P = 2(6 inches) + 2(4 inches)

P = 12 inches + 8 inches

P = 20 inches

Perimeter of Semicircle =
1/2 (π d).

Let PS = Perimeter of Semicircle.

PS = (1/2)(π)(4 inches)

PS = 2π inches.

The wood frame needed to enclose the Norman window is P + PS.

Answer to the wood frame part of the question is 20 inches + 2π inches.

Right? Wrong? Is there an easier method for solving this problem?
 

Dr.Peterson

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Area of Norman window is
24 (inches)^2 + 2π (inches)^2.

Is this correct?
Yes.

How much wood frame is needed to enclose the window?

Let P = perimeter of rectangle part of window.

P = 2l + 2w

P = 2(6 inches) + 2(4 inches)

...

The wood frame needed to enclose the Norman window is P + PS.

Answer to the wood frame part of the question is 20 inches + 2π inches.

Right? Wrong? Is there an easier method for solving this problem?
Do you need to include all four sides of the rectangle in the perimeter of the window?

This is in part an interpretation issue. As I understand it, you are putting a frame around the entire shape, not around each piece.
 

lev888

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Perimeter of a semicircle shape is not half the length of a circle.
More importantly, what exactly is the overall shape? A semicircle and a rectangle framed separately? Or a combined shape that does not need framing along the common edge? That affects what constitutes the perimeter.
 

harpazo

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Yes.


Do you need to include all four sides of the rectangle in the perimeter of the window?

This is in part an interpretation issue. As I understand it, you are putting a frame around the entire shape, not around each piece.
A frame around the entire window.
 

harpazo

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Perimeter of a semicircle shape is not half the length of a circle.
More importantly, what exactly is the overall shape? A semicircle and a rectangle framed separately? Or a combined shape that does not need framing along the common edge? That affects what constitutes the perimeter.
A Norman window has a rectangle connected to a Semicircle.
 

harpazo

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Perimeter of a semicircle shape is not half the length of a circle.
More importantly, what exactly is the overall shape? A semicircle and a rectangle framed separately? Or a combined shape that does not need framing along the common edge? That affects what constitutes the perimeter.
You said:

"Perimeter of a semicircle shape is not half the length of a circle."

I found the formula online. Now, is my answer correct or not?
 

MarkFL

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If I recall correctly, you made the same mistake the last time you posted this question, where you included all 4 sides of the rectangle in the perimeter. :(
 

harpazo

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If I recall correctly, you made the same mistake the last time you posted this question, where you included all 4 sides of the rectangle in the perimeter. :(
How much wood frame is needed to enclose the window?

Let P = perimeter of rectangle part of window.

P = 2l + 2w

P = 2(6 inches) + 4 inches

P = 16 inches

Perimeter of Semicircle =
1/2 (π d).

Let PS = Perimeter of Semicircle.

PS = (1/2)(π)(4 inches)

PS = 2π inches.

The wood frame needed to enclose the Norman window is P + PS.

Answer to the wood frame part of the question is 16 inches + 2π inches.
 

lev888

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The answer is correct. But I still don't agree with your choice of words.

Perimeter - the continuous line forming the boundary of a closed geometric figure.
What exactly do you mean by semicircle? Closed shape? Then its perimeter is not 1/2 (π d). One dimensional arc? Then it does NOT have a perimeter. It has length.
Similar problem with the rectangle.
You wrote: "P = 2l + 2w, P = 2(6 inches) + 4 inches"
How did 2w become 4?
If you are looking at 3 sides of the rectangle the term perimeter no longer applies.
 

harpazo

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The answer is correct. But I still don't agree with your choice of words.

Perimeter - the continuous line forming the boundary of a closed geometric figure.
What exactly do you mean by semicircle? Closed shape? Then its perimeter is not 1/2 (π d). One dimensional arc? Then it does NOT have a perimeter. It has length.
Similar problem with the rectangle.
You wrote: "P = 2l + 2w, P = 2(6 inches) + 4 inches"
How did 2w become 4?
If you are looking at 3 sides of the rectangle the term perimeter no longer applies.
Different words but found the right answer.
 

lev888

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Different words but found the right answer.
Good luck with your studies. Going to use the convenient "Ignore" button on your profile.
 

harpazo

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Otis

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… Going to use the convenient "Ignore" button on your profile.
At the moment, I'm thinking about joining you …

\(\;\)
 

harpazo

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Otis

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Increase the army, right?
The army is imaginary, harpazo. It exists only in your mind. Those people you've talked about before, your family and friends, and the others you've mentioned around New York and from your past, are not against you. The same is true about the people you've argued with at various math forums. Nobody is out to get you. Your mind is where the problem is.

\(\;\)
 

JeffM

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Harpazo

One thing math should have taught you is to be careful in your use of words. "Perimeter" refers to the length of the boundary of a closed figure; it is not s synonym for "length" in general. People who fuss at you about vocabulary are trying to teach you an important aspect of math, where words are very carefully defined.
 

harpazo

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Harpazo

One thing math should have taught you is to be careful in your use of words. "Perimeter" refers to the length of the boundary of a closed figure; it is not s synonym for "length" in general. People who fuss at you about vocabulary are trying to teach you an important aspect of math, where words are very carefully defined.
I post my questions exactly as typed in the textbook. I make no alterations. Of course the correct meaning of words is super important not only in math but in every subject.
 

Otis

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I post my questions exactly as typed in the textbook … I make no alterations …
No, you do not type exercises word-for-word. Some parts are accurate, and other parts you've paraphrased or described vaguely, instead. You also omit stuff that's important to the exercise, sometimes (like the instructions, for example).

And, really, Jeff and the others are trying to talk about your interpretation of certain math vocabulary (not how you choose to post it). Too often, you're not willing to listen. Your mind is closed, when it comes to constructive criticism.

:(
 

Dr.Peterson

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I post my questions exactly as typed in the textbook. I make no alterations. Of course the correct meaning of words is super important not only in math but in every subject.
The use of words that is being discussed is not in your copying of the problem! Look back at post #9, where it was all explained. I'll repeat the ideas here.

This is what you said:
How much wood frame is needed to enclose the window?

Let P = perimeter of rectangle part of window.
P = 2l + 2w
P = 2(6 inches) + 4 inches
P = 16 inches

Perimeter of Semicircle = 1/2 (π d).
Let PS = Perimeter of Semicircle.
PS = (1/2)(π)(4 inches)
PS = 2π inches.

The wood frame needed to enclose the Norman window is P + PS.
Answer to the wood frame part of the question is 16 inches + 2π inches.
There's nothing technically wrong with your use of the word "perimeter" of the rectangle; the formula you use does give the perimeter. But then you don't actually follow that formula! What you are finding, when you use 4 in place of 2w, is the sum of the three sides of the rectangle that are part of the perimeter of the window. That's exactly right; you just didn't clearly state what you were doing. (If you were a teacher explaining this problem to a student, you would have a very confused student at this point!)

As for the semicircle, there is a technical error; a semicircle is a figure enclosed by an arc and a diameter, and its perimeter is the sum of the two. So rather than "perimeter of semicircle", you should have said "length of arc" or perhaps "arc length of semicircle". Nothing other than the word is wrong here.

I prefer not to talk at all about perimeters of parts of a figure, because those perimeters don't add up to make the perimeter of the whole figure (as areas would). Rather, I would just say that the perimeter of the figure is the sum of two vertical sides (6 in each), one horizontal side (4 in), and an arc (2π in). This is much cleaner, and communicates the concepts better. And what communicates better to others also communicates better to yourself, helping you to think correctly. That, of course, is your goal in practicing word problems, right? So recommendations about correct wording are for your good, not criticisms to defend yourself against.

By the way, assuming you quoted the problem fully and exactly, and there was no picture (I suspect there was), the problem is poorly stated, as it talked only about the length and width (long and short sides) of the rectangle, without indicating clearly which is vertical. I'm pretty sure you've got it right, though, because I've seen pictures of this sort of problem many times, and the long side is typically vertical.
 

JeffM

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I post my questions exactly as typed in the textbook. I make no alterations. Of course the correct meaning of words is super important not only in math but in every subject.
THAT is not the point. The problem, if quoted accurately, did NOT use the word "perimeter." You introduced that word on your own and thereby misled yourself by grabbing two formulas before determining whether they were appropriate. You then rejected lev's comments and lost yourself a more than competent tutor.

People are here to help students, not to harass you. Teaching occurs best by asking challenging questions and pushing people to think.
 
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