Question

A piece of rubber tubing maintains a cylindrical shape as it is stretched. At the instant that the inner radius of the tube is 2 millimeters and the height is 20 millimeters, the inne radius is decreasing at the rate of 0.1 millimeter per second and the height is increasing at the rate of 3 millimeters per second. Which of the following statements about the volume of the tube is true at this instant? (The volume V of a cylinder with radius s and height h is V = Πr²h.) A. The volume is increasing by 41 cubic millimeters per second. B. The volume is decreasing by 47 cubic millimeters per second. C. The volume is increasing by 205 cubic millimeters per second. D. The volume is decreasing by 205 cubic millimeters per second.

Answer 1

The volume of the tube is increasing at a rate of approximately 12.57 cubic millimeters per second. All given options are incorrect; the correct rate of change does not match any of the provided answers.

To determine the rate at which the volume of the rubber tubing is changing, we need to differentiate the formula for volume with respect to time and plug in the given rates of change for the inner radius and height.

The volume of a cylinder is given by the formula
`V = \pi r^2h`.

The rate of change of volume with respect to time is given by the derivative
`(dV)/(dt) = \pi * (2rh * (dr)/(dt) + r^2 * (dh)/(dt).`

At the instant of interest, we have r = 2 mm, h = 20 mm, dr/dt = -0.1 mm/s (since the radius is decreasing), and dh/dt = 3 mm/s.

Substituting these values into the derivative formula gives us:

`(dV)/(dt) = \pi * (2 * 2mm * 20mm * (-0.1mm/s) + (2mm)^2 * 3mm/s).`

Calculate the value:

`(dV)/(dt) = \pi * (-8mm^3/s + 12mm^3/s) = \pi * 4mm^3/s.`

Convert cubic millimeters to cubic millimeters per second:

`4mm^3/s * \pi \approx 12.57mm^3/s.`

Since the result is positive, the volume of the tube is increasing at the rate of approximately 12.57 cubic millimeters per second, which means all given options A, B, C, and D are incorrect.