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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:
2. The region enclosed by the curve with equation
, the x-axis and the lines x=2 and x=4 is rotated through 360º 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
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
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
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 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.