the cylinder and the sphere below have the same radius and the same volume which is 6m

1 answer

To find the radius of the cylinder and the sphere, we'll use the formulas for the volume of each shape.

  1. Volume of a cylinder:
    The volume \( V \) of a cylinder is given by the formula: \[ V = \pi r^2 h \] where \( r \) is the radius and \( h \) is the height.

  2. Volume of a sphere:
    The volume \( V \) of a sphere is given by the formula: \[ V = \frac{4}{3} \pi r^3 \]

Since both shapes have the same volume of 6 m³, we can set up the equations. Let's equate the volumes:

For the Cylinder

Assuming the height of the cylinder is \( h \): \[ \pi r^2 h = 6 \]

For the Sphere

Using the volume formula for the sphere: \[ \frac{4}{3} \pi r^3 = 6 \]

Solving for Radius (r)

From the Sphere's volume: \[ \frac{4}{3} \pi r^3 = 6 \] Multiply both sides by \( \frac{3}{4\pi} \): \[ r^3 = \frac{6 \times 3}{4\pi} = \frac{18}{4\pi} = \frac{9}{2\pi} \] Now take the cube root: \[ r = \left(\frac{9}{2\pi}\right)^{\frac{1}{3}} \]

Finding the height of the Cylinder:

Substituting \( r \) back into the cylinder’s volume equation: \[ \pi r^2 h = 6 \] First, calculate \( r^2 \): \[ r^2 = \left(\frac{9}{2\pi}\right)^{\frac{2}{3}} = \frac{9^{\frac{2}{3}}}{(2\pi)^{\frac{2}{3}}} \] Now substitute \( r^2 \) into the cylinder's volume equation: \[ \pi \frac{9^{\frac{2}{3}}}{(2\pi)^{\frac{2}{3}}} h = 6 \] This simplifies to: \[ \frac{9^{\frac{2}{3}} h}{(2\pi)^{\frac{2}{3}}} = 6 \] Now solve for \( h \): \[ h = 6 \frac{(2\pi)^{\frac{2}{3}}}{9^{\frac{2}{3}}} \] You can compute \( r \) and \( h \) using a calculator for numerical approximations if needed.

This gives a general method for finding the radius of the shapes. If you need the specific numeric values or further assistance with calculations, let me know!