A potter forms a lump of clay into a cylinder. As you wind it up, the length, L, of the cylinder increases and the radius, r, decreases. If the length of the cylinder is increasing by 0.8 cm per second, find the rate at which the radius is changing when the radius is 2 cm and the length is 11 cm. (Of units ).

To find the rate at which the radius is changing when the radius is 2 cm and the length is 11 cm, we can use the relationship between the length (L), radius (r), and volume (V) of a cylinder: V = πr^2L Differentiating both sides of the equation with respect to time (t), we get: dV/dt = 2πrL(dr/dt) + πr^2(dL/dt) We are given that dL/dt = 0.8 cm/s and we need to find dr/dt when r = 2 cm and L = 11 cm. Substituting the known values into the equation and rearranging to solve for dr/dt: 0 = 2π(2)(11)(dr/dt) + π(2^2)(0.8) 0 = 44π(dr/dt) + 4π(0.8) 0 = 44π(dr/dt) + 3.2π Simplifying further: 44π(dr/dt) = -3.2π dr/dt = (-3.2π) / (44π) dr/dt = -0.073 cm/s Therefore, when the radius is 2 cm and the length is 11 cm, the rate at which the radius is changing is approximately -0.073 cm/s. Note that the negative sign indicates that the radius is decreasing with respect to time