Chapter 11, Problem 12PT

To determine

To calculate: The area of land was required for given planetarium.

Answer to Problem 12PT

The area of the planetarium dome base is $602.32{\text{km}}^{\text{2}}$ .

Explanation of Solution

Given information: The base of the dome have a circumference of $\text{87}\text{km}$

Concept Used:

The circumference $C$ of the circle having radius $r$ is

  $C=2\pi r$

The area $A$ of the circle having radius $r$ is

  $A=\pi r^2$

Calculation:

The circumference of dome base is $\text{C}=\text{87}\text{km}$

The formula for the circumference of the circle is

  $C=2\pi r$

By equating the value of C

  $\begin{array}{l}2\pi r=87\\ 2\times \frac{22}{7}\times r=87\\ r=87\times \frac{7}{22}\times \frac{1}{2}\\ r=13.846\text{km}\end{array}$

The radius of the dome base is $r=13.846\text{km}$

Now,

The area of the dome base is calculated as

  $A=\pi r^2$

Put the value of radius in A

  $\begin{array}{l}$ A=\pi \times {(13.846)}^2$$ A=\frac{22}{7}\times {(13.846)}^2$A=602.32{\text{km}}^{\text{2}}\end{array}$

Conclusion:

The area of the planetarium dome base is $602.32{\text{km}}^{\text{2}}$ .