SOLUTION: 1. Oil is dripping from a pipe at a constant rate and forms a circular pool. The area of the pool is increasing at 15cm^2/s. Find, to 3 significant figures, the rate of increase of

Question 163286: 1. Oil is dripping from a pipe at a constant rate and forms a circular pool. The area of the pool is increasing at 15cm^2/s. Find, to 3 significant figures, the rate of increase of the radius of the pool when the area is 50cm^2.
2. The region enclosed by the curve with equation y^2=16x, the x-axis and the lines x=2 and x=4 is rotated through 360� about the x-axis. Find, in terms of π, the volume of the solid generated.
3. A particle P moves in a straight line. At time t seconds, the displacement, s metres, of P from a fixed point O of the line is given by s=2tcost+t^2. Find, in m/s to 3 significant figures, the velocity of P when t=3.
Answer by Edwin McCravy(20060) (Show Source): You can put this solution on YOUR website!

1. Oil is dripping from a pipe at a constant rate and forms a circular pool. The area of the pool is increasing at 15cm^2/s. Find, to 3 significant figures, the rate of increase of the radius of the pool when the area is 50cm^2.






>>...The area of the pool is increasing at 15cm^2/s...<<

That says .

So we substitute that and we have:



But we also have to substitute  when 

So we have to calculate  from 
when  to find out what  is then.






So we substitute that in:











Answer: AMP Parsing Error of [cm/sec]: Invalid function '': opening bracket expected at /home/ichudov/project_locations/algebra.com/templates/Algebra/Expression.pm line 70.
.

2. The region enclosed by the curve with equation , the x-axis and the lines x=2 and x=4 is rotated through 360� about the x-axis. Find, in terms of π, the volume of the solid generated.


First we draw the graph of the parabola .

Taking square roots, we see this is really two graphs 
 and 


 
Next we'll draw in the vertical lines  and :



Now we'll erase everything that is not involved
in the rotation about the x-axis. That leaves only the 
graph of  between  and 
and the x-axis.



We draw a slender rectangle as an element of area

.

Label the top point of the element (x,y),
and the height of it y: 



The formula for the volume of a vertically rotated function
using the disk method is:



The height of that tiny rectangle is y and its width
is dx.

It is the height of that rectangle that will rotate 
about the x-axis, so the radius of rotation is y. The 
leftmost value of x is 2 and the rightmost value of x 
is 4.



Then we replace y by

3. A particle P moves in a straight line. At time t seconds, the displacement, s metres, of P from a fixed point O of the line is given by . Find, in m/s to 3 significant figures, the velocity of P when t=3.


The velocity of P is the derivative of the displacement s with
respect to time t, that is, .






When 



When calculating that be sure your calculator 
is in radian mode, not degree mode.



Explanation of the negative sign:

Suppose the line on which P is moving is horizontal.
If a positive velocity means that P is moving to the
right, then a negative velocity means that P is moving
to the left.  So this negative velocity only indicates
that at the exact instant when 3 seconds have passed,
P is moving left.   

Edwin

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