Question #1.) Gravel is being dumped from a conveyor belt at a rate of 40 ft^3 /
ID: 3288718 • Letter: Q
Question
Question #1.) Gravel is being dumped from a conveyor beltat a rate of 40 ft^3/min and its coarseness issuch that it forms a pile in the shape of a conewhose base diameter and height are alwaysequal. --How fast is the height of the pile increasingwhen the pile is 5 ft high?
/Question #2.)-Two cyclists leave simultaneously from theMath Department at UT. One travels northat 16 mph, the other travels east at 30 mph.Determine the rate at which the distance between them is changing after 2 hours of riding.
Explanation / Answer
Volume of cone = (pi/3)r^2 * h
the problem says that r = h, so rewrite Volume:
V = (pi/3)r^2 * r = (pi/3)r^3
Gravel is added to the pile, thus its volume changes with time:
dV/dt = + 40 ft^3 /min
When h = 5 ft, r= 14ft, and (dh/dt) = (dr/dt) so let's take the derivative of Volume with respect to time:
dV/dt = pi*r^2 (dr/dt)
40 = pi*5^2 * (dr/dt)
(dr/dt) = 40/25pi
=1.6 ft/min
(dh//dt) = (dr/dt) =1.6 ft/min