Step 1 :The surface area of a cylinder is given by the formula \(A = 2\pi rh + 2\pi r^2\). We are given that the surface area is decreasing at a rate of -220 square yards per second, and the height is increasing at a rate of 4 yards per second. We want to find the rate at which the radius is changing.
Step 2 :We can express the rate of change of the surface area with respect to time as \(\frac{dA}{dt} = 2\pi r\frac{dh}{dt} + 2\pi h\frac{dr}{dt} + 4\pi r\frac{dr}{dt}\).
Step 3 :Substituting the given values, we get \(-220 = 2\pi(11)(4) + 2\pi(30)\frac{dr}{dt} + 4\pi(11)\frac{dr}{dt}\).
Step 4 :Solving for \(\frac{dr}{dt}\), we get \(\frac{dr}{dt} = \frac{-220 - 2\pi(11)(4)}{2\pi(30) + 4\pi(11)}\).
Step 5 :Calculating the above expression, we get \(\frac{dr}{dt} = -0.293\) yards per second.
Step 6 :So, the radius of the cylinder is decreasing at a rate of \(\boxed{0.293}\) yards per second.