What is height of the cone only?

Question

The vase company designs a new vase that is shaped like a cylinder on bottom and a cone on top. The catalog states that the width is 12 cm and the total height is 42 cm. What would the height of the cylinder part have to be in order for the total volume to be 3843.36 cubic centimeters?

Answer 1

The volume of a cone is (1/3)π(r^2)h and the volume of a cylinder is π(r^2)h. We know that the volume of the cylinder plus the volume of the cone must equal 3843.36 cubic centimeters. We also know that the radius of the cone and cylinder are equal, r=6 cm, since they are the same width.The final piece of information we were given was the height of the lamp.

 

Unknowns:

 

height of cone: a

height of cylinder: b

 

Since we have two unknowns, we need two equations to solve.

 

1. (1/3)π(r^2)a + π(r^2)b = 3843.36   <--- Equation sums the volume of each shape
2. a + b = 42   <--- Equation sums the heights of each shape

 

From equation 2. we know that b = 42 - a. We can substitute this into equation 1. to find:

 

• (1/3)π(r^2)a + π(r^2)(42 - a) = 3843.36
• π(r^2)[(1/3)a + 42 - a] = 3843.36   <--- Since π(r^2) is in each term we can factor it out.
• 36π[42 - (2/3)a] = 3843.36   <--- For simplicity 6 cm was substituted for r.
• 42 - (2/3)a = 33.98   <--- Divide both sides by 36π
• -(2/3)a = -8.02   <--- Subtract both sides by 42
• a = 12.03

 

We can now substitute the value for a back into equation 2. to get b = 29.97. Therefore the height of the cone is 12.03 cm and the height of the cylinder is 29.97 cm.