(note that we used the product rule for the middle term on the left hand side)
now the rest is algebra. solve for
we have , we proceed by the chain rule2. Differentiate
note that the last factor is the derivative of by the quotient rule
the volume, , of a sphere with radius is given by3. A spherical balloon is being inflated. If the volume of the balloon is increasing at a rate of , how fast is the radius increasing when the radius is 3m?
differentiating implicitly with respect to time, we have
we are told and we want , so plug these in and solve for
yes, draw a diagram. note that it makes a right triangle, the hypotenuse is the length of the rope from the pulley to the boat, call this length , the base is the distance of the boat from the dock, call this distance . The height is 1. Now we are told that (note that it is negative because the distance is decreasing) and we want when4. A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1m higher than the bow of the boat. If the boat is pulled in at a rate of 0.8 m/s, how fast does the boat approach the dock when it is 10m from the dock?
By Pythagoras' Theorem, we have that
now plug in everything on the right to find . note that you can find the value of from equation (1)