Problem:
A building is in the form of a cylinder surmounted by a hemisphere vaulted dome and contains 880/21m^3 of air. If the internal diameter of the dome is equal to the total height above the floor, find the height of the building.

Solution:

Let's break down the problem into smaller parts and solve them step by step.

Step 1: Finding the Volume of the Building

To find the height of the building, we first need to calculate the volume of the entire structure.

The building consists of a cylinder and a hemisphere. The volume of a cylinder can be calculated using the formula:

V_cylinder = πr^2h

where r is the radius of the cylinder and h is the height of the cylinder.

The volume of a hemisphere can be calculated using the formula:

V_hemisphere = (2/3)πr^3

where r is the radius of the hemisphere.

Since the internal diameter of the dome is equal to the total height above the floor, we can conclude that the radius of the cylinder is equal to the radius of the hemisphere.

Let's assume the radius of the cylinder and hemisphere is 'r' and the height of the cylinder is 'h'.

The volume of the building can be calculated by adding the volumes of the cylinder and hemisphere:

V_building = V_cylinder + V_hemisphere
= πr^2h + (2/3)πr^3

Given that V_building = 880/21m^3, we can set up the equation:

880/21 = πr^2h + (2/3)πr^3

Step 2: Simplifying the Equation

To simplify the equation, let's substitute the value of π with 22/7.

880/21 = (22/7)r^2h + (2/3)(22/7)r^3

Simplify further by canceling out common factors:

880/21 = (22/7)(r^2h + (4/3)r^3)

Multiply both sides by 21 to eliminate the fraction:

880 = (22/7)(21)(r^2h + (4/3)r^3)

Cancel out the common factors:

880 = 22(3)(r^2h + (4/3)r^3)

Divide both sides by 22(3):

40 = r^2h + (4/3)r^3

Step 3: Solving for Height (h)

We need one more equation to solve for the height (h) of the building. Since the internal diameter of the dome is equal to the total height above the floor, we can write:

h = 2r

Substitute this value in the equation:

40 = r^2(2r) + (4/3)r^3

Simplify the equation:

40 = 2r^3 + (4/3)r^3

Combine like terms:

40 = (2 + 4/3)r^3

Multiply both sides by 3 to eliminate the fraction:

120 = (6 + 4)r^3

120 = 10r^3

Divide both sides by 10:

12 = r^3

Take the cube root of both sides:

r = ∛12