Exercise 13 Page 602

We want to draw and label two cones with different dimensions but the same volume. Let r_1 and r_2 be their radii. Also, h_1 and h_2 will be the heights of the corresponding cones. We can see this in the graph.

Now, let's remember the formula for the volume of a cone. In the formula r is the radius of the base of the cone, and h is its height. V = 1/3π r^2 hWe can use this formula to write the volumes of the cones V_1 and V_2. V_1 &= 1/3π r_1^2 h_1 V_2 &= 1/3π r_2^2 h_2 For the volumes to be equal, the following equation needs to be true. V_1 &= V_2 1/3π r_1^2 h_1 &= 1/3π r_2^2 h_2 Let's transform the last equality further. This means dividing both sides by 13π. 1/3π r_1^2 h_1 &= 1/3π r_2^2 h_2 & ⇓ r_1^2 h_1 &= r_2^2 h_2 We see that for the volumes to be equal the last equality needs to be true. V_1=V_2 ⇔ r_1^2 h_1 = r_2^2 h_2 There are infinitely many pairs of numbers (r_1,h_1) and (r_2,h_2) that satisfy the equation. Let's then narrow it down to natural numbers. We can start by choosing the radii, for example r_1= 6 and r_2= 3. Let's substitute them into the equation and simplify. This will give us an equation for h_1 and h_2. ( 6)^2 h_1 = ( 3)^2 h_2 ⇓ 36h_1=9h_2 Now we need to pick such values of h_1 and h_2 that the equation will be true. One of the possible combinations is h_1=2 and h_2=8. We can check that it satisfies the equation.

This gives us the dimensions of our cones. We can add them onto the graph.

Note that this is just a sample solution.