Question

A company that manufactures custom aluminum windows makes an eyebrow window that is placed on top of a rectangular window. When a customer orders an eyebrow window, the customer gives the width w and height h of the eyebrow. To make the window, the shop needs to know the radius of the circular arc and the length of the circular arc.

a) If the width and height of the eyebrow are 36 in. and 10 in., respectively, then what is the radius of the circular arc?

b) Find the length of the circular arc for a width of 36 in. and a height of 10 in.

Answer 1

c) Find a formula that expresses the radius of the circular arc r in terms of the width of the eyebrow w and the height of the eyebrow h.

d) Find a formula that expresses the length of the circular arc L in terms of w and h.

Answer 2

Show your work. Express your thoughts. Demonstrate what you can do. Go!

Answer 3

The answers are supposed to be:
a) 21.2
b) 43
c) ((4h^2)+(w^2)/8h)
d) ((w^h)/(h^2)+(0.25w^2)) (((h^2)+(25w^2)/h) arcsin (wh)/(h^2)+(0.25w^2))

Answer 4

I just don't know how to get them..

Answer 5

I guess eyebrow window is like a segment of the circle, with may or may not be a semi circle, now u can do it

Answer 6

I know that the window is a rectangle with a semi circle on top, but I still can't figure out how I can find the answer

Answer 7

Solve for r.

Answer 8

Thank you! I found a!

Answer 9

You should not be trying to "get them". Most of "them" are wrong. You should be trying to solve the problem and find which is correct. There are a few situations where looking at the possible answers is a good thing to do up front. Very few.

Answer 10

They are the correct answers

Answer 11

would you like me to show you how they are done?

Answer 12

Please! :)

Answer 13

Lets do b

The formula for arc length is s = rθ

where s is the arc length and θ is the angle subtended by the arc.
We know r so we just have to work out θ

Answer 14

ok, is the r we're using 21.2?

Answer 15

Yes r - 21.2

Now the angle at the centre of the circle is θ/2

so sin(θ/2) = 18/r and θ/2 = arcsin(18/r)

From here you can find θ in radians

Answer 16

I got 116.2 for theta

Answer 17

no thats in degrees. we need radians

Answer 18

Oh! I found b, so theta is 2.028 and then I multiplied that by 21.2, and got 43! thank you!

Answer 19

yes, well done.

now for c

Answer 20

ok

Answer 21

Using pythagoras we have r^2=(r-h)^2+(w/2)^2
=> r^2 = r^2-2hr+h^2+(w/2)^2
=> 2hr = h^2 + w^2/4

can you see that so far?

Answer 22

I got h+w^2/8

Answer 23

I mean h/2 + (w^2)/8h

Answer 24

yes that's right! and if you put that on a common denominator of 8h you will get your original answer

Answer 25

happy with that?

Answer 26

can you show me how I can get my original answer?

Answer 27

r = h/2 + w^2/8h
=> r = (4h^2 + w^2)/8h

Answer 28

oh, I got it now. Thanks!

Answer 29

Now, I just need help with D

Answer 30

I got a slightly different expression for D

Answer 31

Your D answer is wrong

Using L = rθ and sinθ/2 = w/2r

=> θ = 2arcsin(w/2r)
=> L = 2r.arcsin(w/2r)

Now we substitute our previous expression for r
Follow so far?

Answer 32

would it be this?

(((8h^2)+(2w^2))/(16h))arcsin (w)/(((8h^2)+(2w^2))/(16h))

Answer 33

No, we get this........

L = [(4h^2 + w^2)/4h].arcsin[(4hw)/(4h^2 +w^2)]

Answer 34

Thanks, I got it!