the height of a cylinder is increasing at a rate of 4 inches per second, while the radius is decreasing at a…

Explanation:

Step1: Differentiate volume formula

Given $V = \pi r^{2}h$, using the product - rule $\frac{d(uv)}{dt}=u\frac{dv}{dt}+v\frac{du}{dt}$, we have $\frac{dV}{dt}=\pi\left(2rh\frac{dr}{dt}+r^{2}\frac{dh}{dt}\right)$.

Step2: Identify given values

We know that $\frac{dh}{dt}=4$ in/sec, $\frac{dr}{dt}=- 3$ in/sec, $h = 34$ in, and $r = 15$ in.

Step3: Substitute values

Substitute into $\frac{dV}{dt}=\pi\left(2rh\frac{dr}{dt}+r^{2}\frac{dh}{dt}\right)$: [ \begin{align*} \frac{dV}{dt}&=\pi\left(2\times15\times34\times(-3)+15^{2}\times4\right)\ &=\pi\left(-30\times34\times3 + 225\times4\right)\ &=\pi\left(-3060+900\right)\ &=\pi\times(- 2160)\ &\approx - 6785.8 \end{align*} ]

Answer:

$-6785.8$