the height of a cylinder is increasing at a rate of 2 inches per second, while the radius is decreasing at a…

Explanation:

Step1: Differentiate volume formula

The volume of a cylinder is $V = \pi r^{2}h$. Using the product - rule $\frac{d(uv)}{dt}=u\frac{dv}{dt}+v\frac{du}{dt}$, where $u = \pi r^{2}$ and $v = h$. So $\frac{dV}{dt}=\pi\left(2rh\frac{dr}{dt}+r^{2}\frac{dh}{dt}\right)$.

Step2: Substitute given values

We know that $\frac{dh}{dt}=2$ in/sec, $\frac{dr}{dt}=- 4$ in/sec, $h = 50$ in, and $r = 11$ in. Substitute these values into the formula for $\frac{dV}{dt}$: [ \begin{align*} \frac{dV}{dt}&=\pi\left[2\times11\times50\times(-4)+11^{2}\times2\right]\ &=\pi\left[ - 4400+242\right]\ &=\pi\times(-4158)\ &\approx - 13053.2 \end{align*} ]

Answer:

$-13053.2$