the height and radius of a cylinder are both changing, resulting in a change in the cylinders volume. the…

the height and radius of a cylinder are both changing, resulting in a change in the cylinders volume. the height of the cylinder is increasing at a constant rate of 2 meters per second, and the volume is decreasing at a constant rate of 800 cubic meters per second. at the instant when the radius of the cylinder is 5 meters and the volume is 648 cubic meters, what is the rate of change in the radius? the volume of a cylinder can be found with the equation v = πr²h. round your answer to three decimal places.

the height and radius of a cylinder are both changing, resulting in a change in the cylinders volume. the height of the cylinder is increasing at a constant rate of 2 meters per second, and the volume is decreasing at a constant rate of 800 cubic meters per second. at the instant when the radius of the cylinder is 5 meters and the volume is 648 cubic meters, what is the rate of change in the radius? the volume of a cylinder can be found with the equation v = πr²h. round your answer to three decimal places.

Answer

Explanation:

Step1: Differentiate volume formula

Differentiate $V = \pi r^{2}h$ with respect to time $t$ using the product - rule. The product - rule states that if $y = uv$, then $\frac{dy}{dt}=u\frac{dv}{dt}+v\frac{du}{dt}$. Here, $u=\pi r^{2}$ and $v = h$, so $\frac{dV}{dt}=\pi r^{2}\frac{dh}{dt}+2\pi rh\frac{dr}{dt}$.

Step2: Find the height

We know that $V=\pi r^{2}h$. Given $V = 648$ and $r = 5$, we can solve for $h$: $h=\frac{V}{\pi r^{2}}=\frac{648}{\pi\times5^{2}}=\frac{648}{25\pi}$.

Step3: Substitute known values

We are given that $\frac{dh}{dt}=2$ and $\frac{dV}{dt}=- 800$, $r = 5$, and $h=\frac{648}{25\pi}$. Substitute these values into $\frac{dV}{dt}=\pi r^{2}\frac{dh}{dt}+2\pi rh\frac{dr}{dt}$: $-800=\pi\times5^{2}\times2+2\pi\times5\times\frac{648}{25\pi}\times\frac{dr}{dt}$.

Step4: Simplify the equation

First, simplify the left - hand side terms: $\pi\times5^{2}\times2 = 50\pi\approx50\times3.14159 = 157.0795$. $2\pi\times5\times\frac{648}{25\pi}=\frac{2\times5\times648}{25}=\frac{6480}{25}=259.2$. The equation becomes $-800 = 157.0795+259.2\times\frac{dr}{dt}$.

Step5: Solve for $\frac{dr}{dt}$

Rearrange the equation to solve for $\frac{dr}{dt}$: $259.2\times\frac{dr}{dt}=-800 - 157.0795=-957.0795$. $\frac{dr}{dt}=\frac{-957.0795}{259.2}\approx - 3.692$.

Answer:

$-3.692$