We know that the arc length is .
A sector is made up of 2 radii and this length, so . Solve this for .
I am currently studying for my calculus 1 final (which is on the 12th) and have reached some questions I cannot answer. For those who might have the book they are in section 8.2.8 of "Modern Engineering Mathematics 4th Edition" by Glyn James.
Question #17 States:
A wire of length L metres is bent so as to form the boundary of a sector of a circle of radius r metres and angle x radians. Show that
and prove that the area of the of the sector is greatest when the radius is L/4.
The second part of the question is not too difficult, I have managed to work through it, but I don't know how to approach the first part of the question. I have tried relating the circumference, using triangles, using the sector length formula, and other unsuccessful methods to derive that formula.
Help would be appreciated, thanks.
I see, I was thinking about it wrong. I interpreted the length as just the arc length and not the perimeter. Thanks, that solves my problem.
I am new here, is it bad form to ask a similar (concept) problem in the same thread?
Alright, Question 2 is from the same set of questions:
Gas escapes from a spherical balloon at 2m^3.min^-1. How fast us the surface area shrinking when the radius is 12m.
I worked through this and attained an answer of 1/6m^2.min^-1 which is half the actual answer of 1/3m^2.min^-1.
I re-arranged the volume formula to give me radius then subbed that into the surface area formula, differentiated (I ended up with SA'=2/((3v)/(4pi))^1/3) ), subbed in 12 and solved. The problem is that I don't know how to incorporate the gas escape rate into the calculation.
We're told that , so we need somehow to bring time into this.
You probably know (and if not, you can calculate by differentiating the volume, as you're probably aware), that for a sphere, where S is the surface area.
We need to find .
. The problem becomes finding
We know that and so .
We also know that .
Now we can use the fact that to give:
And then you can just substitute in to finish.
Edit: The negative comes from the fact that the balloon is being depleted/shrinking; if it had been expanding, I would have used a positive. Either way, the result is the magnitude so it isn't relevant here.