Function expression, area of a window

[LogarithmLuke]

Homework Statement

http://www.inter-ped.no/skei/MAT1013%20Matematikk%201T%20%20bokmal.pdf Page 10 task 4 for a picture.
The window above consists of a rectangle and a semicircle. The windows perimeter is 8.0m.
What does the radius in the semicircle have to be for the area of the window to be as big as possible? Calculate this area.

Homework Equations

Perimeter circle:2*pi*R Are circle: Pi*rsquared

The Attempt at a Solution

So the total perimeter of the window has to be Pi*r+2h+2L=8. The problem here is that we need a function for the area expressed with just R. Since we don't know any lengths i don't know how to remove the other variables in the equation.

 

[Mark44]

Homework Statement

http://www.inter-ped.no/skei/MAT1013%20Matematikk%201T%20%20bokmal.pdf Page 10 task 4 for a picture.
The window above consists of a rectangle and a semicircle. The windows perimeter is 8.0m.
What does the radius in the semicircle have to be for the area of the window to be as big as possible? Calculate this area.

Homework Equations

Perimeter circle:2*pi*R Are circle: Pi*rsquared

The Attempt at a Solution

So the total perimeter of the window has to be Pi*r+2h+2L=8. The problem here is that we need a function for the area expressed with just R. Since we don't know any lengths i don't know how to remove the other variables in the equation.

You don't need three variables in your equation for the perimeter. The diameter of the semicircular part at the top is the same as the width of the window.
What is the total area of the window? That's what you have to make as large as possible.

 

[AntSC]

Why is there a variable 'L' in your perimeter equation? Can you simplify it?
Why don't you write your perimeter equation in terms of 'h' and 'r' and then eliminate 'h'.

 

[LogarithmLuke]

How do you know that the width of the window is the same as the diameter of the semicircle?

Theres both a length and a width in the rectangle.

 

[SteamKing]

How do you know that the width of the window is the same as the diameter of the semicircle?

Theres both a length and a width in the rectangle.

Because if the width of the window is not the same as the diameter of the semicircle, then you've got notches where the semicircle and the rectangle meet, adding more complication to the formula for the perimeter of the window.

The window should look like one of these:

 

[LogarithmLuke]

Okay, then you have w canceled out, what about L?

 

[SteamKing]

Okay, then you have w canceled out, what about L?

It's not clear what you mean by "w canceled out". And it's not clear why there is a variable L either.

The picture on p. 10 of the test booklet shows a window with only two variable dimensions: the radius of the semicircular top and the height of the rectangular lower portion.

These are all the dimensions you need to calculate the perimeter and the area of the window.

What's the formula for the perimeter for a window like the one shown in the picture above? The formula for the area of the window?

 

[LogarithmLuke]

I mean that we have found an expression for width expressed with r. And now we need an expression for the hight, using r or just numbers.

Perimeter would be the sum of the perimeters of the two figures, same concept with the area. The problem now the way i see it is just finding an expression for the height of the rectangle.

 

[SteamKing]

I mean that we have found an expression for width expressed with r. And now we need an expression for the hight, using r or just numbers.

The height of the rectangle is h, as shown in the figure on p. 10 of the test booklet. What you must find is the proportions of r and h which maximize the area of the window while keeping the perimeter constant at 8.0 m.

Perimeter would be the sum of the perimeters of the two figures, same concept with the area. The problem now the way i see it is just finding an expression for the height of the rectangle.

No, the perimeter of the window is not the sum of the perimeters of the semi-circle and the rectangle. A portion of the perimeters of these two figures overlaps, and thus is not included in the perimeter of the window. That's why I asked you to write the formula for the perimeter of the window, to make sure that your understanding and my understanding of the perimeter of the window are the same.

 

[LogarithmLuke]

How would you go about solving the problem then? We haven't learned any window formulas and i am quite sure we aren't meant to use that. What i was thinking was to find a formula for the area of the window, and then derivate it and find the max point.

 

[SteamKing]

How would you go about solving the problem then? We haven't learned any window formulas and i am quite sure we aren't meant to use that. What i was thinking was to find a formula for the area of the window, and then derivate it and find the max point.

It's not clear what you mean by window formulas. All we are talking about here is applying standard Euclidean geometry to analyze a certain geometrical figure which is composed of a semi-circle and a rectangle.

Instead of a window, this could be a piece of land someone is interested in buying, or the shape of a special piece of cloth, or whatever. You are confusing the thing with how to solve the problem.

You still haven't written down the correct formula for the perimeter of this geometric shape, which uses the information provided.

Yes, since you want to find the maximum area of the window, you will have to write down a formula for this area and differentiate (this is the correct term for taking the derivative in English) it to find the values of h and r which maximize the area of the window, while the perimeter remains fixed at 8.0 meters.

In any event, write these things down in your next post:
1. The perimeter of the window, using h for the height of the rectangular portion of the window and r for the radius of the semicircle on top.
2. The area of the window, also using r and h alone.

You will not need to invent any extra variables to solve this problem.

 

[LogarithmLuke]

1 I can't find any other solutions to this than the following:4r+2h+pi*r
2 If there aren't any spesific formulas for a shape like this, in my head it has to be: 4r*h+(pi*r^2)/2
I know you said we can't summarize the perimeters/areas but these are the only solutions i can find.
If we assume these are the correct formulas(which they probably aren't) I am guessing the next step would be to solve the equation for r, since we need the area to be a function of only r.

 

[DrClaude]

1 I can't find any other solutions to this than the following:4r+2h+pi*r

Explain what you are summing up, because this is not correct.

2 If there aren't any spesific formulas for a shape like this, in my head it has to be: 4r*h+pi*r

Again, please explain your reasoning. In particular, is the equation compatible with an area (imagine that r and h have units)?

If we assume these are the correct formulas(which they probably aren't) I am guessing the next step would be to solve the equation for h, since we need the area to be a function of only h.

But the question asks for the radius, so it would be easier to get an equation for r.

 

[LogarithmLuke]

Corrected some things in my post. I was just adding the perimeter and area formulas, i don't see how there can be any other formulas to describe the area and perimeter of this window. I meant that the area has to be a function of r. So i was thinking to express h with r.

If the width is the diameter of the semicircle, or 2r, in my head it has to be 2r+2r(perimeter formula).

 

[SammyS]

Corrected some things in my post. I was just adding the perimeter and area formulas, i don't see how there can be any other formulas to describe the area and perimeter of this window. I meant that the area has to be a function of r. So i was thinking to express h with r.

If the width is the diameter of the semicircle, or 2r, in my head it has to be 2r+2r(perimeter formula).

What is the definition of the perimeter of any two dimensional object in general ? -- not asking for a formula

 

[Mark44]

Corrected some things in my post. I was just adding the perimeter and area formulas, i don't see how there can be any other formulas to describe the area and perimeter of this window.

It seems that you are trying to find and use some cookbook formula for the perimeter of this window instead of using your brain to reason out what such a formula would need to be. The window is made up of basic geometric shapes, so it shouldn't be as difficult as you are making it to derive your own formula for the perimeter of this window.

I meant that the area has to be a function of r. So i was thinking to express h with r.

If h is the height of the rectangular portion of the window, h has nothing to do with r. The width of the window, which I would call w, does involve r, which I already said in post #2.

If the width is the diameter of the semicircle, or 2r, in my head it has to be 2r+2r(perimeter formula).

I dont' know what you're saying here. What does "it" refer to? "In my head it has to be 2r + 2r" What is "it"?

 

[LogarithmLuke]

What is the definition of the perimeter of any two dimensional object in general ? -- not asking for a formula

Just all the sides summarized, if we are thinking about the same thing.

It seems that you are trying to find and use some cookbook formula for the perimeter of this window instead of using your brain to reason out what such a formula would need to be

Like i said, i just summarized the formulas for each figure, i don't see any other way to do it. I'm sorry if I am missing something that should be obvious here, but I am not engineering level mathematician yet, that's why I am asking for help.

I dont' know what you're saying here. What does "it" refer to? "In my head it has to be 2r + 2r" What is "it"?

Just the formula, i meant that the formula has to be that way the way i see it.

 

[Mark44]

Look at the picture that SteamKing posted (post #5). The perimeter of one of these windows is the total of the lengths along the outer edge of the window. How many straight sides are there? How many curved sides are there?

 

[LogarithmLuke]

Oh, i see it now. So there are only three sides in the rectangle. That defenitely changes things.

 

[Mark44]

Oh, i see it now. So there are only three sides in the rectangle. That defenitely changes things.

YES!
Now write an equation with the known perimeter, 8m, and the three sides and half circle. From that equation, you can solve for one of the two unknowns.

Then, write an expression that represents the area of the window, and we can go from there.

 

[LogarithmLuke]

Ok, so 8=pi*r+2h+2r r=(8-2h)/3pi
Im not totally sure about the area since there are only 3 sides. However I am guessing it would be: A=2r*h+(pi*r^2)/2

 

[SammyS]

Ok, so 8=pi*r+2h+2r r=(8-2h)/3pi
I'm not totally sure about the area since there are only 3 sides. However I'm guessing it would be: A=2r*h+(pi*r^2)/2

That area is correct.

The area is simply the sum of the two areas.

The area of the rectangle.
The area of the semicircle.​

In particular, the area of the rectangle does not depend upon which sides are used for the overall perimeter and which side are not used.

Now, since you need the area in terms of r only and not h, you need to solve the perimeter equation for h, not r. (When you solved it for r, your result was incorrect anyway.)

 

[LogarithmLuke]

Is H=(8-2r)/2*pi*r correct?
And r is the squareroot of (8-2h)/2pi ?
So, if the formula for h is correct, next step I am guessing would be to put our expression for h in the perimeter formula, find a number value for h and then put that back into the area formula and it should give us our function for the area of window?

 

[SammyS]

Is H=(8-2r)/2*pi*r correct?

No.

 

[Mark44]

Ok, so 8=pi*r+2h+2r r=(8-2h)/3pi

The first equation above is correct for the perimeter. How did you get the second equation?

Is H=(8-2r)/2*pi*r correct?

No.
Please show your work in solving for h in the equation $8 = \pi r + 2h + 2r$.

And r is the squareroot of (8-2h)/2pi ?
So, if the formula for h is correct, next step I am guessing would be to put our expression for h in the perimeter formula, find a number value for h and then put that back into the area formula and it should give us our function for the area of window?

 

[LogarithmLuke]

I subtracted both sides by 2r, then divided by pi*r, then i divided both sides by 2 to get h alone.

 

[SammyS]

I subtracted both sides by 2r, then divided by pi*r, then i divided both sides by 2 to get h alone.

So that is (step by step):

$\displaystyle\ 8 = \pi r + 2h + 2r \$

$\displaystyle\ 8-2r = \pi r + 2h \$

Now divide by (π r) (not really what you should do)
$\displaystyle\ \frac{8-2r}{\pi r} = \frac{\pi r + 2h }{\pi r} \ \to\ 1+\frac{2h }{\pi r}$

divide both sides by 2
$\displaystyle\ \frac{8-2r}{2\pi r} = \frac 12+\frac{h }{\pi r}$

This doesn't give h.

 

[LogarithmLuke]

Okay, i tried again. From the secound step, i subtracted both sides by pi*r and then divdided by 2 which gave me the following equation: (8-2r-pi*r)/2=h

 

[SammyS]

Okay, i tried again. From the second step, i subtracted both sides by π*r and then divided by 2 which gave me the following equation: (8-2r-π*r)/2=h

Much better.

 

[LogarithmLuke]

I think i can manage the rest now, thanks a lot for all your help guys :)