Question

Boring a Cylinder The mechanics at Lincoln Automotive are reboring a 6 -in. deep cylinder to fit a new piston. The machine they are using increases the cylinder's radius one-thousandth of an inch every 3 min. How rapidly is the cylinder volume increasing when the bore (diameter) is 3.800 in.?

Short answer

The volume of the cylinder is increasing at a rate of 0.0238 cubic inches per minute when the bore is 3.800 inches.

Step-by-step solution

Identifying Given Values

We're given that the height, \(h\), of the cylinder is 6 inches and its radius, \(r\), is increasing at a rate of one-thousandth of an inch every 3 minutes. To express the rate in inches per minute, it will be \(\frac{1}{3000}\) inches per minute. We're also given that, at the moment we're interested in, the bore (diameter) of the cylinder is 3.8 inches, so the radius is \(r = 1.9\) inches.

Setting Up the Relationship

We're asked about the rate of change of volume, \(V\), with respect to time. We know the formula for the volume of a cylinder is \(V = \pi r^2h\). So, taking the derivative with respect to time \(t\) of both sides gives us \(\frac{dV}{dt} = 2\pi r\frac{dr}{dt}h\).

Substituting the Known Values

We can find the rate of change of the volume by substituting \(h = 6\), \(r = 1.9\), and \(\frac{dr}{dt} = \frac{1}{3000}\) into the expression derived, resulting in: \(\frac{dV}{dt} = 2\pi \times 1.9 \times \frac{1}{3000} \times 6 = 0.0238\) cubic inches per minute.