Expert: Scotto - 5/15/2010

Question
The height of the cone is 7.7 cm and the radius is 5.2 cm. As scoops of water were added (1 scoop = 2 teaspoons), the water level rose like so: 1 scoop-2.2 cm, 2 scoops-3.7 cm, 3 scoops-4.2 cm, 4 scoops-4.7 cm, 5 scoops-5.2 cm, and 6 scoops-5.7 cm. a) Write an equation for the volume of the cone as a function of the height. b) The rate of change of thw volume is being held constant at 1 scoop per minute; determine how many cubic cm of water is in one scoop. c) Find the rate of change of height dh/dt when h=one-half of its total height. Show work for all parts.

Answer
It looks more like it is a cone at the bottom and a cylinder at the top,
for it fills 0.5 cm for every scoop after the 1st two.
The 1st fills it 2.2 cm and the 2nd fills it 1.5 cm.

The equation for depth filled is 3.7 cm plus 0.5 for each scoop after that.
This gives h = 3.7 + 0.5(x-2).  The 0.5(x-2) = 0.5x - 1, and that -1 can be combined
with the 3.7, giving 3.7 - 1 = 2.7.  From these, a linear equation for the height can
be derived in the form h = B + Cx.  This would be the equation h = 2.7 + 0.5x for x>=2.
At x=1, the depth is 2.2.  After this, it goes by the equation.

To get the volume, this equation must be multiplied by the surface area.
The radius is siad to be r = 5.2 cm.  The equation to find the surface area is A = πr².

a) The equation for the volume is found by V = Ah,
where A was π*5.2² = 2.704π and h = 2.7 + 0.5x, where x was the scoops.

b) Since the depth is rising by 0.5 cm for each scoop, compute the volume V = πr²h for h = 0.5 cm
with the radius r = 5.2 cm (5.2² = 2.704).  That gives the volume V in cc's of liquid.

c) The rate of change was given in the first question as 0.5.  To do this, write down a table with the scoops in one column and the depth in the next column.  In the next column, the values would be the change in depth.  This would be in the 3rd column.
The table would be
1  2.2
2  3.7  1.5
3  4.2  0.5
4  4.7  0.5
5  5.2  0.5
6  5.7  0.5.

Note that 1.5 = 3.7-2.2, the 1st 0.5 = 4.2-3.7, the next 0.5 = 4.7 -4.2, etc.
Thus, dh/dt = 0.5 cm/min.