Question

A building is in the form of a cylinder surmounted by a hemispherical vaulted dome and contains 10, 51 m³ of air. If the internal diameter of the dome is equal to of the total height above the floor, find the height of the building.

Answer 1

To find the height of the building, we need to calculate the volumes of the cylindrical and hemispherical parts and add them together. The height can be expressed in terms of the radius of the dome. By substituting the values into the volume equations and simplifying, we can solve for h.

Step-by-step explanation:

To find the height of the building, we need to calculate the volume of the cylindrical part and the hemispherical part and then add them together. The volume of a cylinder is given by V = πr^2h, where r is the radius and h is the height. The volume of a hemispherical dome is given by V = (2/3)πr^3.

Let's assume the height of the building is h. The total volume of the building is given as 10, 51 m³.

Using the information that the internal diameter of the dome is equal to the total height above the floor, we can express the radius of the dome in terms of h as r = h/2.

Substituting the values into the volume equations:

• Volume of cylinder: V1 = π(h/2)^2h = (π/4)h³
• Volume of dome: V2 = (2/3)π(h/2)^3 = (1/6)πh³

Adding the two volumes together, we get the equation (π/4)h³ + (1/6)πh³ = 10, 51. Simplifying this equation gives us h³ = (10, 51 × 24)/(π × 5). Taking the cube root of both sides, we can solve for h.