Calculus Notes

Example 2.7.1.

Consider a cylindrical water tower with a leaky valve at the bottom. Water leaks at a rate of \(1\text{ft}^3\text{/min}\text{.}\) The tower has dimensions of: \(r = 20\text{ft}\text{,}\)\(h = 50\text{ft}\text{.}\)

How quickly is the height of water changing at 35ft?

1. assign letters:

   • v = volume of water in tower

   • h = height of water

   • r = radius

   • t = time

2. identify rates:

   • \(\displaystyle \dfrac{dv}{dt} = -1 \text{ft}^3 \text{/min}\)

   • we need to find \(\dfrac{dh}{dt}|_{h = 35}\)

3. equation:

   • \(\displaystyle V = \pi r^2 h\)

4. differentiate:

   • \(\displaystyle \dfrac{d}{dx} \left( V(t) \right) = \dfrac{d}{dt} \left( \pi r^2 h(t) \right) \)

   • \(\displaystyle \dfrac{dv}{dt} = \pi r^2 h(t) \dfrac{dh}{dt} \)

   • \(\displaystyle \dfrac{dh}{dt} = \dfrac{1}{\pi r^2} \cdot \dfrac{dv}{dt} \)

5. substitution:

   • \(\displaystyle \dfrac{dh}{dt}|_{h = 35} = \dfrac{1}{\pi r^2} \cdot (-1) = -0.0008 \text{ft/min} \)