Question

A rocket consists of a right circular cylinder of height 20 m surmounted by a cone whose height and diameter are equal and whose radius is the same as that of the cylindrical section. What should this radius be (correct to two decimal places) if the total volume is to be 500 π/3 m³?

Answer 1

To find the radius for a rocket with a cylindrical and conical section having a total volume of 500 π/3 m³, we use the volume formulas for a cylinder and a cone, set up an equation with the given total volume, and solve for the radius.

Step-by-step explanation:

Calculating the Radius of a Rocket's Cylindrical and Conical Sections

To determine the radius required for a rocket that consists of a right circular cylinder and a cone where the total volume is 500 π/3 m³, we need to set up an equation that includes the volumes of both the cylinder and the cone.

The volume of the cylinder (Vcyl) is given by:

Vcyl = πr²h, where ‘r’ is the radius and ‘h’ is the height of the cylinder.

The volume of the cone (Vcone) is given by:

Vcone = 1/3πr²h, since the height of the cone is equal to its diameter, which is 2r.

For the total volume of the rocket to be 500 π/3 m³, we set up the following equation:

500 π/3 = πr²(20) + 1/3πr²(2r)

Solving for ‘r’, we need to:

1. Combine the terms.
2. Divide both sides by π.
3. Isolate ‘r³’ on one side by subtracting the term containing the height of the cylinder from the total volume.
4. Divide by the coefficient of ‘r³’ to find its value.
5. Take the cube root of ‘r³’ to find the radius ‘r’.

By following these steps, we can calculate the exact radius needed for the rocket to achieve the specified total volume.