Thread: Rate of increase on conical pile

Sand is being dumped at a constant rate onto a conical pile, which is 5 feet high. When you see it again, 1 hour and 10 minutes later, the pile is 10 feet high. At what rate is the height increasing?

If you can help me figure out this problem I'd appreciate it

We may first write all of the information given in mathematical notation:



In order that the problem have an answer, the ratio of the height to the radius, and thus the angle of the tip of the cone, must remain the same. (Otherwise, we could have the cone start out narrow and then get wider as approaches , lowering our answer for dramatically.) Therefore, we may say that



for some , and that



In order to find the value of at , in minutes, we may use the equations above together with the fact that