Mechanics are reboring a 6in deep cylinder to fit a new piston. The machine they are using increases the cylinders radius one-thousandths of an inch every 3 min. How rapidly is the cylinder volume increasing when the bore is 3.800 in.?

To find the rate at which the cylinder volume is increasing, we need to differentiate the volume formula with respect to time and then substitute the given values.

Let's denote:
- r as the radius of the cylinder (in inches)
- h as the height of the cylinder (in inches)
- V as the volume of the cylinder (in cubic inches)

The volume of a cylinder is given by the formula: V = πr^2h

Since we are interested in finding how rapidly the cylinder volume is increasing with time, we need to differentiate both sides of the equation with respect to time (dt):

dV/dt = d/dt (πr^2h)

Let's break down this equation using the product rule and chain rule of differentiation:

dV/dt = 2πrh(dr/dt) + πr^2(dh/dt)

We are given that the machine increases the cylinder's radius by one-thousandths of an inch every 3 minutes:

dr/dt = 0.001 inches/3 minutes = 0.0003333 inches/minute

Now we need to find the height of the cylinder when the bore is 3.800 inches. Since the original depth is 6 inches, and we want to know the height when the bore is 3.800 inches, we subtract the new bore depth from the original depth:

h = 6 inches - 3.800 inches = 2.200 inches

We also need to find the current radius of the cylinder when the bore is 3.800 inches. Since the bore is increasing at a rate of 0.001 inches every 3 minutes, we can calculate the radius:

r = initial radius + (dr/dt) * t
r = 3 inches + (0.0003333 inches/minute) * t

To find the value of t (time) when the bore is 3.800 inches, we substitute r = 3.800 inches into the equation and solve for t:

3.800 inches = 3 inches + (0.0003333 inches/minute) * t

t = (3.800 inches - 3 inches) / (0.0003333 inches/minute)
t ≈ 2397.96 minutes

Now we have all the information we need to calculate the rate at which the cylinder volume is increasing when the bore is 3.800 inches.

Substitute r = 3.800 inches and h = 2.200 inches into the differentiated equation:

dV/dt = 2πrh(dr/dt) + πr^2(dh/dt)
dV/dt = 2π(3.800 inches)(2.200 inches)(0.0003333 inches/minute) + π(3.800 inches)^2(0 inches/minute)

Calculating the above expression will give you the rate at which the cylinder volume is increasing when the bore is 3.800 inches.

v = pi r^2 h

dv/dt = pi(2rh dr/dt + r^2 dh/dt)
now plug in your values to get dv/dt

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