Teacher Guide, Geometry Activity 2
TEKS addressed: (a)(2); (a)(4); (a)(6); (G.1)(B); (G.5)(B)
Vocabulary includes literal equations and delta form of a slope equation.
This makes a good Pre-AP* geometry problem because students must apply many concepts in one problem. Teachers may choose to teach parts of this problem at different points during the year. However, if these problems are presented as a unit students can not skip the hard problems and still do well on the unit. Teachers may choose to do all or just some of these questions. The goal is to incorporate calculus concepts early and often, to give students glimpses of future learning, and to show them how they will use this concept over the next four years.
(a) List in words the items in the problem which DO NOT change.
The dimensions of the cone and the cylinder; the radius of the water in the cylinder
(b) List in words the items in the problem which DO change.
The height and radius of the water in the cone; the height of the water in the cylinder
(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.?
The water drains more slowly at the beginning, so the slopes are flatter. As the water drains, the slopes become steeper because the water drains faster. The graph is decreasing because the slopes are negative and concave down because the tangent lines would be above the curve.
(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.
1:3
(h) If the water in the cone is 6 foot deep, what is the radius of the water?
2 ft.
(i) If the water in the cone is 6 foot deep, what is the surface area of the water?
4π
(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?
1:8
(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?
1:2
(m) What is the relationship between your answers for (h) and (i)?
Volume is the cube of the scale factor for the radius
(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.
The change in volume with respect to time = π / 9 times the height squared times the change in height with respect to time.
(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?
W = πr2h
(v) What is the volume of the cylinder in terms of y, the height of the water?
W = 400πy
(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:
(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.
The water is draining from the conical tank, so the rate of change of volume is negative. The water is draining into the cylindrical tank, so the rate of change of volume will be the opposite of the rate of drainage.
(z) At what rate is y changing when h=3? Write the sentence in symbolic notation, then solve indicating units of measure.
