A potter forms a piece of clay into a cylinder. As he rolls it, the length,L, of the cylinder increases and the radius, r, decreases. If the length of the cylinder is increasing at .1 cm per second, find the rate at which the radius is changing when the radius is 1 cm. and the length is 5 cm.

Volume is remaining the same. Think on that.

V=PI*r^2 * L
dv/dt= 2PI*L r dr/dt+ PI*r^2 dL/dt=0

solve for dr/dt

-.001

To find the rate at which the radius is changing, we can use the formula for the relationship between the radius and length of a cylinder:

V = πr^2L

where V is the volume of the cylinder. Since the length is increasing at 0.1 cm/s, we can differentiate both sides of the equation with respect to time to find the relationship between the rates of change:

dV/dt = 2πrL(dr/dt) + πr^2(dL/dt)

Since we are interested in finding the rate at which the radius is changing when the radius is 1 cm and the length is 5 cm, we can plug in these values into the equation:

dV/dt = 2π(1)(5)(dr/dt) + π(1)^2(0.1)

Simplifying the expression:

dV/dt = 10π(dr/dt) + 0.1π

Since the volume of a cylinder is given by V = πr^2L, we can substitute V = π(1)^2(5) into the equation:

10π(dr/dt) + 0.1π = π(1)^2(5)

Simplifying further gives:

10π(dr/dt) + 0.1π = 5π
10(dr/dt) + 0.1 = 5

Now we can solve for the rate at which the radius is changing:

10(dr/dt) = 5 - 0.1
10(dr/dt) = 4.9
(dr/dt) = 4.9/10
(dr/dt) ≈ 0.49 cm/s

Therefore, when the radius is 1 cm and the length is 5 cm, the rate at which the radius is changing is approximately 0.49 cm/s.

To find the rate at which the radius is changing, we need to use the relationship between the length and the radius of the cylinder, as well as the given information about the changing length. Let's use the formula for the volume of a cylinder to establish this relationship:

Volume, V = πr²L

We can differentiate both sides of this equation with respect to time (t) to relate the changing variables:

dV/dt = d/dt (πr²L)

The left side of the equation represents the rate at which the volume is changing, which we know from the given information (dV/dt = 0.1 cm³/s). We will also need to determine how the other variables change with time, which involves finding expressions for dL/dt (the rate of change of length) and dr/dt (the rate of change of radius).

Let's proceed by differentiating πr²L with respect to time using the product rule:

dV/dt = πr²(dL/dt) + 2πrL(dr/dt)

Since we are interested in finding the rate at which the radius is changing (dr/dt), we can rewrite the equation as:

2πrL(dr/dt) = dV/dt - πr²(dL/dt)

Now, let's substitute the given values for the length and the rate of change of length into the equation, given that L = 5 cm and dL/dt = 0.1 cm/s:

2π(1 cm)(5 cm)(dr/dt) = 0.1 cm³/s - π(1 cm)²(0.1 cm/s)

Simplifying further:

10π(dr/dt) = 0.1 cm³/s - 0.1π cm³/s

Now, we can isolate dr/dt by dividing both sides by 10π:

dr/dt = (0.1 cm³/s - 0.1π cm³/s) / (10π)

Finally, let's evaluate this expression using the value of π and calculate the rate at which the radius is changing when the radius is 1 cm and the length is 5 cm:

dr/dt ≈ (0.1 cm³/s - 0.1π cm³/s) / (10π)
dr/dt ≈ (0.1 cm³/s - 0.31 cm³/s) / (31.42)
dr/dt ≈ -0.21 cm³/s / 31.42
dr/dt ≈ -0.0067 cm/s

Therefore, the rate at which the radius is changing when the radius is 1 cm and the length is 5 cm is approximately -0.0067 cm/s.