A solid cylinder with a radius of 40 cm has the same mass as a solid sphere of radius r

To determine the relationship between the radius of the solid sphere and the solid cylinder, we can start by analyzing their volumes and using the concept of mass and density.

First, let’s calculate the volume of the solid cylinder. The formula for the volume (V) of a cylinder is given by:

V_cylinder = πr²h

For our cylinder, we have a radius (r) of 40 cm. Since the height (h) isn’t provided, we’ll denote it as h. Thus, the volume of the cylinder is:

V_cylinder = π(40 cm)²h = 1600πh cm³

Next, we calculate the volume of the solid sphere using the formula:

V_sphere = (4/3)πr³

Let’s denote the radius of the sphere as r (which we are trying to find). Thus, the volume of the sphere is:

V_sphere = (4/3)πr³ cm³

Since the masses of both objects are equal, we can say:

Density x Volume_cylinder = Density x Volume_sphere

If we assume both shapes have the same density (which is reasonable under the same material assumption), we can simplify this to:

V_cylinder = V_sphere

Substituting our volume expressions, we get:

1600πh = (4/3)πr³

We can divide both sides by π (as long as π is not zero) to simplify:

1600h = (4/3)r³

To find the radius r of the sphere in terms of h, we solve for r:

r³ = (1600h * 3)/4 = 1200h

Taking the cube root of both sides, we get:

r = (1200h)^(1/3)

In conclusion, the radius of the solid sphere (r) can be expressed in terms of the height (h) of the solid cylinder. Thus, knowing the height of the cylinder allows you to find the radius of the sphere having the same mass.