You have the correct radius of the water's surface at the time the depth of the water is 3 ft.
Can you show your working for the problem, so we may identify where you are erring?
I have tried to solve the following problem. When I perform the differentiation before supplying the given variables, I am left with two unknowns: dr/dt and dh/dt. When I try to remove a variable by supplying the radius as a constant before differentiation, I receive the wrong answer (it is exactly 3x too big). Several other problems later on give me the same trouble, so I am hoping that an understanding of this problem will enlighten me to the proper method to solve the others.
"A water tank is in the shape of a cone with vertical axis and vertex downward. The tank's radius is 3 ft. and the tank is 5 ft. high. At first the tank is full of water, but at time t = 0 (in seconds), a small hole at the vertex is opened, and the water begins to drain. When the height of water in the tank has dropped to 3 ft, the water is flowing out at 0.02 ft^3/s. At what rate, in feet per second, is the water level dropping then?"
I am working with a radius of 9/5 ft of the surface of the water.
Thank you in advance!
Ok. The knowns: h_{0}=5 ft. r_{0}=3 ft, h_{1}=3 ft, dV/dt = -0.02 ft^{3}/s. Formula for the volume of a cone: V = 1/3 pi r^2 h. Because the h_{1} is 3/5 of h_{0}, r_{1} = 9/5.
Sought variable: dh/dt at t_{1}
Now, one of the cardinal rules for doing this type of problem is that we don't insert known quantities until after differentiation. So my first attempt went like this:
V = 1/3 pi r^2 h.
dV/dt = (((2pi rh)/3) dr/dt) + (((pi r^2)/3)dh/dt)
-0.02 = (18pi/5)dr/dt + 81pi/75 dh/dt.
This leaves me with two unknowns: dr/dt and dh/dt.
My second approach was to supply the radius as a constant before differentiation, to leave just one unknown: dh/dt.
So:
V = 1/3 pi r^2 h.
V = 1/3 pi (81/25) h
dV/Dt = 1/3 pi (81/25) dh/dt
-0.02 = 1/3 pi (81/25) dh/dt
dh/dt = ((3)(-0.02)(25))/81pi
= -1.5/81 pi
= -1/54pi ft./s.
But this answer is exactly three times two big according to the answer key. The answer key has -1/162 pi. What did I do wrong?