SOLUTION: I am struggling to understand how to work this problem. It is a function word problem for the volume of water.
Water flowing into a conical drinking cup with an altitude of 4 i
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Question 1123579: I am struggling to understand how to work this problem. It is a function word problem for the volume of water.
Water flowing into a conical drinking cup with an altitude of 4 inches and a radius of 2 inches. write the volume V of the water as a function of its depth h.
Found 2 solutions by greenestamps, solver91311:
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!
As the water flows into the conical cup, the surface of the water remains flat.
That means the cones of water at different times are all similar.
Since the radius of the whole cup is half the height of the whole cup, the radius of the surface of the water at any time is half the depth of the water at that time. So at all times r = (h/2).
The formula for the volume of a cone is
Substitute r = h/2 in that formula to get a function for the volume of water as a function of the depth of the water.
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
.
The illustration is a cross-section of your cone through the axis. Note that the altitude of the cone and a radius of the base form a triangle, and the depth of the water and the radius of the surface of the water form a similar triangle, similar being used in the formal sense so that the corresponding sides of the two triangles are proportional.
Hence, 
from which we can easily derive a relationship between 
and 
:
Considering the formula for the volume of a cone of radius and height :
\ =\ \frac{1}{3}\pi{r^2}h)
and the relationship between and just established, we can make the substitution of 
for .
\ =\ \frac{1}{3}\pi{\(\frac{h}{2}\)^2}h)
Which simplifies to:
I will leave the rest as an exercise for the student.
John
My calculator said it, I believe it, that settles it
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