a piston is seated at the top of a cylindrical chamber with radius 5 cm when it starts moving into the…

a piston is seated at the top of a cylindrical chamber with radius 5 cm when it starts moving into the chamber at a constant speed of 6 cm/s (see figure). what is the rate of change of the volume of the cylinder when the piston is 11 cm from the base of the chamber? when the piston is 11 cm from the base of the chamber, the volume of the cylinder is changing at a rate of about (round to the nearest hundredth as needed.)

a piston is seated at the top of a cylindrical chamber with radius 5 cm when it starts moving into the chamber at a constant speed of 6 cm/s (see figure). what is the rate of change of the volume of the cylinder when the piston is 11 cm from the base of the chamber? when the piston is 11 cm from the base of the chamber, the volume of the cylinder is changing at a rate of about (round to the nearest hundredth as needed.)

Answer

Explanation:

Step1: Recall the volume formula for a cylinder

The volume formula for a cylinder is (V=\pi r^{2}h). Here, the radius (r = 5) cm (constant), and (h) is the height (distance of the piston from the base of the chamber). So (V=\pi\times(5)^{2}\times h=25\pi h).

Step2: Differentiate the volume with respect to time

Differentiate (V = 25\pi h) with respect to time (t) using the chain - rule. (\frac{dV}{dt}=25\pi\frac{dh}{dt}). We are given that (\frac{dh}{dt}=6) cm/s (the speed of the piston).

Step3: Calculate the rate of change of volume

Substitute (\frac{dh}{dt}=6) into the equation (\frac{dV}{dt}=25\pi\frac{dh}{dt}). (\frac{dV}{dt}=25\pi\times6). [ \begin{align*} \frac{dV}{dt}&=150\pi\ &\approx150\times 3.14159\ & = 471.24 \end{align*} ]

Answer:

(471.24) cm³/s