We want to draw and label two with different dimensions but the same . Let r1 and r2 be their . Also, h1 and h2 will be the heights of the corresponding cones. We can see this in the graph.
Now, let's remember the formula for the . In the formula
r is the radius of the base of the cone, and
h is its height.
V=31πr2h
We can use this formula to write the volumes of the cones
V1 and
V2.
V1V2=31πr12h1=31πr22h2
For the volumes to be equal, the following equation needs to be true.
V131πr12h1=V2=31πr22h2
Let's transform the last equality further. This means dividing both sides by
31π.
31πr12h1r12h1=31πr22h2⇓=r22h2
We see that for the volumes to be equal the last equality needs to be true.
V1=V2⇔r12h1=r22h2
There are infinitely many pairs of numbers
(r1,h1) and
(r2,h2) that satisfy the equation. Let's then narrow it down to . We can start by choosing the radii, for example
r1=6 and
r2=3. Let's substitute them into the equation and simplify. This will give us an equation for
h1 and
h2.
(6)2h1=(3)2h2⇓36h1=9h2
Now we need to pick such values of
h1 and
h2 that the equation will be true. One of the possible combinations is
h1=2 and
h2=8. We can check that it satisfies the equation.
36h1=9h2
36(2)=9(8)
72=72✓
This gives us the dimensions of our cones. We can add them onto the graph.
Note that this is just a sample solution.