To determine the population density of the antelopes around the watering hole, we need to follow these steps:
1. Calculate the Area of the Circle:
We know the radius of the circle is [tex]$\frac{3}{4}$[/tex] km. The formula for the area of a circle is given by:
[tex]\[
\text{Area} = \pi \times r^2
\][/tex]
Given that [tex]$\pi$[/tex] is 3.14 and the radius [tex]\( r \)[/tex] is [tex]$\frac{3}{4}$[/tex] km, first find [tex]\( r^2 \)[/tex]:
[tex]\[
r^2 = \left(\frac{3}{4}\right)^2 = \frac{9}{16}
\][/tex]
Now, multiply this by [tex]$\pi$[/tex] to find the area:
[tex]\[
\text{Area} = 3.14 \times \frac{9}{16}
\][/tex]
[tex]\[
\text{Area} \approx 3.14 \times 0.5625 = 1.76625 \; \text{km}^2
\][/tex]
2. Calculate the Population Density:
Population density is defined as the number of antelopes per square kilometer. We have 32 antelopes in the area we calculated.
Using the formula for population density:
[tex]\[
\text{Population Density} = \frac{\text{Number of Antelopes}}{\text{Area}}
\][/tex]
[tex]\[
\text{Population Density} = \frac{32}{1.76625}
\][/tex]
[tex]\[
\text{Population Density} \approx 18.117480537862704 \; \text{antelopes per} \; \text{km}^2
\][/tex]
3. Round the Population Density:
Finally, we need to round this value to the nearest whole number:
[tex]\[
\text{Population Density} \approx 18 \; \text{antelopes per} \; \text{km}^2
\][/tex]
So, the population density of the antelopes near the watering hole is approximately 18 antelopes per square kilometer (rounded to the nearest whole number).
