Question

Archimedes (ca. 287-212 B.C.) was able to use clever geometric means to determine the relative volumes of a cylinder and the cone and paraboloid that would fit snugly into it (1800 years before Newton and Leibniz). With calculus, you don't have to be a genius to reach the same conclusions.

Determine the ratio of the volume of a (right circular) cone to the volume of a cylinder with the same height and base radius: (First you will need to determine the equation of the line which rotates to generate the cone.)

Answer 1

Answer:

The ration of the volume of  cone to that of   cylinder is   $\frac{V_{cone}}{V_{cy}} = \frac{1}{3}$  

Step-by-step explanation:

From the question we are told that

 The volume of a cone is mathematically represented as

        $V_{cone} = \frac{1}{3} \pi r^2 h$

The volume of a cylinder is mathematically represented as

       $V_{cy} = \pi r^2 h$

Now the ratio we are to obtain is

          $\frac{V_{cone}}{V_{cy}} = \frac{\frac{1}{3} \pi r^2 h}{\pi r^2 h}$    

                   $\frac{V_{cone}}{V_{cy}} = \frac{\frac{1}{3} }{1}$         Note: this is possible because the height and base

                   $\frac{V_{cone}}{V_{cy}} = \frac{1}{3}$        radius  are the same

Answer 2

By taking the quotient between the volume of a cone and the volume of a cylinder, we will see that the ratio is 1/3.

Finding the ratio between the volumes:

We know that for a cylinder of radius R and height H the volume is:

V = pi*R^2*H

While for a right cone with a base of radius R and a height H, the volume is:

V' = pi*R^2*H/3

You can see that these are only different by a factor of (1/3).

Taking the quotient we get:

V'/V = ( pi*R^2*H)/(3*pi*R^2*H) = 1/3

So the ratio of the volume of a right circular cone to the volume of a cylinder is:

r = 1/3

If you want to learn more about volumes, you can read:

https://brainly.com/question/8994737