Glencoe Math: Course 3, Volume 2
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Glencoe Math: Course 3, Volume 2 View details
2. Volume of Cones
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Exercise 13 Page 602

Think about pairs of numbers representing the radius and the height that give the same volume.

Example Solution:

Practice makes perfect

We want to draw and label two cones with different dimensions but the same volume. Let and be their radii. Also, and will be the heights of the corresponding cones. We can see this in the graph.

A green cone with height h one and base radius r one, accompanied by an orange cone with height h two and base radius r two
Now, let's remember the formula for the volume of a cone. In the formula is the radius of the base of the cone, and is its height.
We can use this formula to write the volumes of the cones and
For the volumes to be equal, the following equation needs to be true.
Let's transform the last equality further. This means dividing both sides by
We see that for the volumes to be equal the last equality needs to be true.
There are infinitely many pairs of numbers and that satisfy the equation. Let's then narrow it down to natural numbers. We can start by choosing the radii, for example and Let's substitute them into the equation and simplify. This will give us an equation for and
Now we need to pick such values of and that the equation will be true. One of the possible combinations is and We can check that it satisfies the equation.
This gives us the dimensions of our cones. We can add them onto the graph.
Cones

Note that this is just a sample solution.