Student Worksheet, Geometry Activity 2
As shown in the figure below, water is draining from a conical tank with height 12 feet and diameter 8 feet into a cylindrical tank that has a base with area of 400π square feet. The depth, h, in feet, of the water in the conical tank is changing at the rate of (h-12) feet per minute.
(a) List in words the items in the problem which DO NOT change.
(b) List in words the items in the problem which DO change.
(c) Write the formula for volume of a cone.
(d) Find the capacity of the cone.
(e) Sketch a graph of height versus time as the water drains from the cone. Describe the slopes of the graph. Is the graph increasing, decreasing, concave up, concave down, etc.?
(f) If the water is draining from the cone at a rate of 9 ft.³ per hour, how long would it take to empty the cone?
(g) Find the ratio of the radius of the water to the depth of the water in the cone.
(h) If the water in the cone is 6 foot deep, what is the radius of the water?
(i) If the water in the cone is 6 foot deep, what is the surface area of the water?
(j) If the water in the cone is 6 foot deep, what is the volume of the water in the cone?
(k) If the water in the cone is 6 foot deep, what is the ratio of the volume of the water in the cone to the capacity of the cone?
(l) If the water in the cone is 6 foot deep, what is the ratio of the radius of the water to the radius of the cone?
(m) What is the relationship between your answers for (h) and (i)?
(n) Find the radius of the water in the cone as a function of its height.
(o) Find the volume of the water in the cone as a function of the height.
(p) Write a symbol to describe the change in height with respect to change in time using the delta form of a slope equation. Using a "d" for "delta" (Δ), rewrite the equation to represent an instantaneous rate of change.
(q) It can be determined that 
Write the meaning of these symbols in words.
(r) The depth h, in feet, of the water in the cylindrical tank is changing at a rate of h-12 feet per minute. Write the statement in symbolic notation.
(s) Rewrite the equation for the change in volume with respect to time using part (q).
(t) At what rate is the volume of the water in the conical tank changing when h= 3? Write the sentence with symbols, then determine the rate. Indicate units.
(u) What is the formula for volume of a cylinder using W for volume?
(v) What is the volume of the cylinder in terms of y, the height of the water?
(w) If the water in the cone is 6 foot deep when it begins to drain, what will be the height of the water in the cylinder when the cone is empty (assuming the cylinder starts off empty)?
(x) The change in the volume of the cylindrical tank can be described by the following:
Write the meaning of the symbols in words, then solve algebraically for 
(y) Describe the difference in what is happening to the water in the conical tank and what is happening to the water in the cylindrical tank.
(z) At what rate is y changing when h=3? Write the sentence in symbolic notation, then solve indicating units of measure.
