I am trying to solve the following problem:
https://docs.google.com/document/d/1pibMkkuY_1Onve8WzUBRhxF7_qZlgLyiFAOpjTEnjcI/edit?usp=sharing
This is my attempt at solving:
Given: Cone: r=2cm.; h=5cm.; dh/dt = 3cm/s @ the point of submersion; Cylinder: r=7cm.; h = L <--- Depth of the Water.
Unknown: dL/dt @ the point of submersion
I started off by stating that the volume of the water in the cylinder at the point of submersion is equal to the volume of the water - (pi)(r^2)(L) minus the volume of the cone - (pi/3)(r^2)(h).
After substituting the given values, I found that the volume of the cylinder is 49pi(L) - (pi/3)(h). When I take the derivative of both in respect to t on both sides, I found that the derivative of volume is equal to 0 because it's rate of change is constant throughout the point of submersion. This results in: 49pi(dL/dt) = (pi/3)(dh/dt). By using the given dh/dt, I found that the depth of the water is increasing at 1/49 cm/s at the point of submersion.
Question:
Is my setup correct, and is my solution sound? Is the overall height of the cylinder important to consider in solving this problem?