Question

A proposed design for a habitable Mars colony is a hemispherical
biodome used to maintain a breathable atmosphere for the colonists.
Estimate the radius of the biodome if it is required to contain 5.5
billion cubic feet of air.

Answer 1

Answer:

r = 1.37 billion feet

Step-by-step explanation:

It is given that,

A proposed design for a habitable Mars colony is a hemispherical  biodome used to maintain a breathable atmosphere for the colonists.

The volume of biodome is 5.5  billion cubic feet of air.

It is in the shape of a hemisphere. The volume of a hemisphere is given by :

$V=\dfrac{2}{3}\pi r^3$, r = radius

V = 5.5 billion cubic feet

$r^3=\dfrac{3V}{2\pi}\\\\r^3=\dfrac{3\times 5.5}{2\times 3.14}\\\\r=(\dfrac{3\times 5.5}{2\times 3.14})^{1/3}\\\\r=1.37\ \text{billion feet}$

So, the radius of the biodome is 1.37 billion feet.

Answer 2

The radius of the biodome if it is required to contain 5.5  billion cubic feet of air is 1.38 billion feet and this can be determined by using the formula of the volume of the hemisphere.

Given :

• A proposed design for a habitable Mars colony is a hemispherical  biodome used to maintain a breathable atmosphere for the colonists.
• Volume of the biodome is 5.5 billion cubic feet.

The following steps can be used in order to determine the radius of the biodome:

Step 1 - The formula of the volume of the hemisphere can be used in order to determine the radius of the biodome.

Step 2 - Formula of the volume of the hemisphere is given by:

$\rm V =\dfrac{2}{3}\pi r^3$

where r is the radius of the hemisphere.

Step 3 - Now, substitute the value of V in the above formula in order to determine the value of 'r'.

$\rm 5.5=\dfrac{2}{3}\pi r^3$

Step 4 - Simplify the above expression.

$r^3=\dfrac{5.5\times 3}{2\pi}$

$r = \sqrt[3]{\dfrac{16.5}{2\pi}}$

r = 1.38 billion feet

For more information, refer to the link given below:

https://brainly.com/question/15861918