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as i am new to calculus & probability,pls show how to do these problems(may be very trivial)
problem 1:show that the height of a right circular cylinder of the greatest volume that can be inscribed in a right circular cone is one-third of that of the cone.
(using derivatives and limits....)
problem 2:using integration,find the area enclosed by the curves and
problem 3: out of 21 tickets marked with numbers from 1 to 21, three are drawn at random.Find the probability that the numbers on them are in A.P.
please help!!!thanks in advance
1) i think my diagram is correct. if so, take a relationship between h and r from H and R which are constants
then calculate the volume V of the cylinder and differentiate it with respect to either h (or r). Find for which value for h (or r) you get the
take . If for each calculated value for h, you'll get a minus for it means V is maximum for that value for that h
Let the height of the cone be "h" and the radius of the base "R". If you set up a coordinate system so that the origin is at the center of the base and the axis is along the positive z-axis, then the slant side, in the xz-plane, passes through (0, 0, h) and (R, 0, 0) and so has equation z= h(1- x/R), y= 0.
A cylinder of radius r, with center at (0, 0, 0) and axis along the positive z-axis, inscribed in that cone will go up to that line at x= r; its height is h(1- r/R) and so its volume is [itex]\pi r^2(h(1- r/R))= \pi h (r^2- r^3/R)[/itex]. That will have maximum volume when its derivative is 0- [itex]\pi r^2h (2r- 3r^2/R)= 0[/itex]. [itex]r(2- 3r/)= 0[/itex] when r= 0 (the minimum volume) or r= (2/3)R.
Now, the height is given by h(1- r/R)= h(1- 2/3)= (1/3)h.
Hello, earthboy!
Problem 3 is intriguing.
There are: .Problem 3: Out of 21 tickets marked with numbers from 1 to 21, tjree are
drawn at random. Find the probability that the numbers on them are in A.P.
How many of them are in Arithmetic Progression?
I tried to derive a formula, and came up with this:
. . With integers from 1 to ,
. . . .
Hence, for AP's.
Therefore: .