A building is in the form of a cylinder surmounted by a hemispherical vaylted dome and contains 1144/21 m3 of air. If the internal diameter of the dome is equal to 4/5th of the total height above the floor, find the height of the building

v = 2/3 pi r^3 + pi r^2 h

d = 4/5 h, so r = 2h/5
but that's the total height. So, in terms of wall height, r = 2h/3

= pi r^2 (2r/3 + h)
1144/21 = pi (2h/3)^2 (2/3 * 2h/3 + h) = 52pi/81 h^3

h = 3 m

so, the wall is 3m high, and the dome is another 2m high, making the whole structure 5m high.

To find the height of the building, we need to break down the problem and find the volume of the building.

Let's start by considering the cylinder part of the building. The volume of a cylinder is calculated using the formula:

V_cylinder = π * r^2 * h_cylinder

where:
- V_cylinder is the volume of the cylinder
- π is a mathematical constant approximately equal to 3.14159
- r is the radius of the cylinder (which is half of the diameter)
- h_cylinder is the height of the cylinder

Next, let's consider the hemispherical vaulted dome on top of the cylinder. The volume of a hemisphere is given by the formula:

V_hemisphere = (2/3) * π * r^3

But since the dome is cut in half to form a vault, we only need half of the volume:

V_dome = (1/2) * V_hemisphere = (1/2) * (2/3) * π * r^3 = (1/3) * π * r^3

Now, we can calculate the total volume of the building by summing up the volumes of the cylinder and the dome:

V_total = V_cylinder + V_dome

According to the problem, the volume of air in the building is given as 1144/21 m^3. So we have:

V_total = 1144/21

Now, let's substitute the formulas for V_cylinder and V_dome:

π * r^2 * h_cylinder + (1/3) * π * r^3 = 1144/21

We are also given that the internal diameter of the dome is equal to 4/5th of the total height above the floor. We can express this relationship as:

diameter = (4/5) * h_total

But the diameter is equal to twice the radius, so we have:

2r = (4/5) * h_total

Now, we can substitute this expression for r in terms of h_total into the volume equation:

π * ((2/5) * h_total)^2 * h_cylinder + (1/3) * π * ((2/5) * h_total)^3 = 1144/21

Rearranging and simplifying the equation, we get:

(16/25) * π * h_total^3 + (4/25) * π * h_total^3 = 1144/21

Combining like terms further, we have:

(20/25) * π * h_total^3 = 1144/21

Simplifying the left-hand side and cross-multiplying, we obtain:

(4/5) * π * h_total^3 = 1144/21

Now, we can solve for h_total by isolating it:

h_total^3 = (1144/21) * (5/4) * (1/π)

Finally, we can find the cube root of both sides to get the value of h_total:

h_total = (cube root of [(1144/21) * (5/4) * (1/π)])

After calculating the right-hand side of the equation, you will find the value of h_total, which represents the height of the building.