EMAT 6600: Problem Solving in Mathematics
Dr. J. Wilson, Instructor
Ryan FoxÕs Solution to the Cone Half Full Problem
Problem: Consider a right circular cone with diameter of 12 cm for the base and altitude of 16 cm. If the cone is held with its vertex down and water placed into it until half full, what is the depth of the water?
Solution Path: I will begin by first finding the volume of the full cone. This will give us the following calculations.
To find the volume of the water, which is a cone that is half full, we will take half of the volume we just found: 96¹. We will assume that the ratio of the radius of the cone to the height of the cone is consistent for any value of the radius and height. In this particular cone, if the radius is 6 cm and the height is 16 cm, then we can say that for any radius measurement r and any height measurement h. We can take the volume 96¹ and solve for the radius of the water in the cone; once we find the radius of the cone, we will be able to find out the height of the water in the cone.
From our calculations, we know that if the depth of the water is 8 times the cube root of 4, or approximately 12.699, centimeters, the amount of water in the cone equals half the volume of the entire cone.
We can verify the volume of the water is half the total volume of the cone by making the appropriate substitution.
Problem (Extension): Find the depth x when a cone of radius r and height h is half full of water? Use this general solution to find the solution for this particular cone.
Solution Path: From the previous problem, we will assume that the height is some constant multiple of the radius: , where m and n are non-zero integers. We know that the volume of the water must equal half the volume of the cone. From this, we can say . Since the ratio of height and radius is maintained throughout the cone, then we can also say that . Making the appropriate substitutions and calculations we can arrive at our desired result.
For the previous problem, we know that the radius is of length 6 and the length of the height is 8/3 times the length of the radius. Making the appropriate substitutions in the previous equation, we can verify the value of this expression.
Problem: Let k range from 0 to 1. Plot the depth of water x as k goes from 0 to 1 for this cone. That is, find x = f(k) and graph it.
We are now interested in finding the depth of water for any fraction of the volume of the cone. We will let k equal the fraction of the cone that we want filled with water and we will let x represent the depth of water in the cone. Using our previous solution, we can say that the value of k equals the value of 1/2 from our previous solution. Therefore, the depth of the water can be expressed in terms of the fraction of the cone that is to be filled with water and the given radius of the base of the cone. For this particular problem, we know that the relationship between the height of the cone and the length of the radius of the base of the cone equals 8/3. Making the appropriate substitutions, we see that the depth of the water in the cone equals
which can be graphed in the k-x axis, as seen in the following representation from the Graphing Calculator software program
The red vertical line is included to represent the maximum fraction of the cone that can be filled with water and the blue horizontal line is included to represent the maximum depth of water in the cone. As we can see, the maximum depth of the water does indeed occur when the cone is full of water.