Question

A space telescope on a mountaintop is housed inside of a cylindrical building with a hemispheric dome. If the circumference of the dome is 84 feet, and the total height of the building up to the top of the dome is 91 feet, what is the approximate total volume of the building? *

Answer 1

Answer:

 $185553.7\text{ feet}^3$

Step-by-step explanation:

GIVEN: A space telescope on a mountaintop is housed inside of a cylindrical building with a hemispheric dome. If the circumference of the dome is $84\text{ feet}$, and the total height of the building up to the top of the dome is $91\text{ feet}$.

TO FIND: what is the approximate total volume of the building.

SOLUTION:

let the height of the mountaintop be $h\text{ feet}$

As the dome hemispherical.

circumference of a hemisphere $=\frac{1}{2}\times2\pi\times radius$

                                                    $=\pi\times radius$

                                                    $\frac{22}{7}\times radius=84$

                                                    $radius=26.75\text{ feet}$

total height of the building up to the top of the dome $=\text{radius of hemisphere}+\text{height of mountaintop}$

$=26.75+h=91$

$h=64.25\text{ feet}$

Volume of building $=\text{volume of cylindrical mountaintop}+\text{volume of dome}$

                                $=\pi(radius)^2h+\frac{2}{3}\pi(radius)^3$

as radius of mountain top is same as dome

putting values

                                $=3.14(26.75)^264.75+\frac{2}{3}3.14(26.75)^3$      

                               $=145484.6+40069.1$

                               $185553.7\text{ feet}^3$

Hence the total volume of the building is $185553.7\text{ feet}^3$