Math Is Fun Forum

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Curved surface area

Can anyone help me with this question please?

The diagram shows a cross-section of a tunnel.
It shows a segment of a circle of centre O and radius 5m.
AB is a chord of the circle.
AB = 8m
The length of the tunnel is 40m.

Calculate the curved surface area of the tunnel to 3.s.f.

Thanks!

Last edited by Daniel123 (2007-05-29 02:56:47)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Curved surface area

First calculate the obtuse angle AOB. You can either split the isosceles triangle AOB into two congruent halves, or apply the cosine rule to ΔAOB. You get (in radians)

The reflex angle AOB is therefore

The curved surface area of a cylinder with radius 5 m and length 40 m is 2π(5)(40) = 400π m[sup]2[/sup]. So if A is the curved surface area of the tunnel,

Last edited by JaneFairfax (2007-05-29 04:06:50)

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: Curved surface area

Thanks Jane...although im not 100% sure of whats going on... firstly ive never used radians (as im only GCSE level). Then you said "The curved surface area of a cylinder with radius 5 m and length 40 m is π(5)²40".. isnt that the volume?... lol i also dont understand why you have divided A by 1000π... and why you need to know the size of the reflex angle? sorry...

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: Curved surface area

Wowoah! I got some crazy answer! It's probably wrong, but let me know if it isn't!

Let C be the midpoint of AB.

Hence,

and so,

Hence, the area is inside the curved part

The area inside the triangular part

So the total area of the cross section is:

The circumference of the cross section

So the total surface area is:

≈ 1364.43 = 1360 m² (3.s.f)

Last edited by Identity (2007-05-29 04:02:21)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Curved surface area

Thanks Jane...although im not 100% sure of whats going on... firstly ive never used radians (as im only GCSE level). Then you said "The curved surface area of a cylinder with radius 5 m and length 40 m is π(5)²40".. isnt that the volume?... lol i also dont understand why you have divided A by 1000π... and why you need to know the size of the reflex angle? sorry...

You need to use the reflex angle because the curved cross section of the tunnel is the major arc of a circle.

And you are right. I made a mistake and used a volume formula instead of the area formula! Yikes! I have gone back and edited my post now.

And if you want to use degrees, you can. The formula for A then beocomes

Last edited by JaneFairfax (2007-05-29 04:15:59)

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Curved surface area

Identity's given the full surface area of the tunnel, which includes the area of the ends and the floor. If you only want the curved bit, then it's:

Edit: Post collision. Two different ways of finding it though, so not completely useless.

---

Why did the vector cross the road?
It wanted to be normal.

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: Curved surface area

Thanks all of you... just wondering.. could it be done by calculating the curved perimeter of the major segment and multiplying by 40? .. because isnt the curved surface area like a rectangle anyway?

Last edited by Daniel123 (2007-05-29 08:11:29)

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Curved surface area

Yes, you can. That's pretty much what we were doing.

---

Why did the vector cross the road?
It wanted to be normal.

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: Curved surface area

Ohh right lol... wasnt 100% following what you were doing as you may have guessed. Just so i know... with LATEX... how do you do a multiplication sign? and a negative power? thanks

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Curved surface area

You do a multiplication sign by writing "\times".
As you know already, you can do simple powers as normal, eg. "x^2".

If the power contains more than one symbol though, you need to encase it in curly brackets so that the computer knows what you're talking about. eg. "x^{-k}".

For future reference, this forum has an extremely useful feature in that you can click on a LaTeX image to be able to see the code that made it (minus the [math] tags).

---

Why did the vector cross the road?
It wanted to be normal.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Curved surface area

Also, the degree symbol is:

^\circ

(which is actually a superscript circle).
­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: Curved surface area

Thanks... i have edited by calculations! Im also wondering... how do you get the letters to look like normal letters - like you have for 'm'. I took mathsy's advice and looked at the code, but i didnt understand what was going on!

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Curved surface area

You can use either \mbox{} or \mathrm{}.

10m^2

produces

Instead, try

10\mathrm{m}^2 or 10\mbox{m}^2

which will give

(both the same). To make it look even better, insert a small space \, between the 0 and the m:

10\,\mathrm{m}^2

=

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: Curved surface area