Final answer:
To find the population density of Las Pythagoras, we calculate the area of the city, which is in the shape of an isosceles right triangle with legs of 25 km, and then divide the population of 100,000 by that area. The population density is approximately 320 people per square kilometer.
Step-by-step explanation:
The student is asking for the population density of a city named Las Pythagoras, which is shaped as an isosceles right triangle. The legs of this triangle are each 25 kilometers long. To find the population density, we need to calculate the area of the triangle and then divide the population by this area.
First, we apply the Pythagorean theorem to find the area of the triangle:
\( a^2 + b^2 = c^2 \) where \( a \) and \( b \) are the legs and \( c \) is the hypotenuse.
Given that \( a = 25 \) km and \( b = 25 \) km, we have:
\( 25^2 + 25^2 = c^2 \)
\( 625 + 625 = c^2 \)
\( 1250 = c^2 \)
\( c = \sqrt{1250} \)
\( c \approx 35.36 \) km
Now, we calculate the area of the triangle using the legs (since it's a right triangle):
Area = \( \frac{1}{2} \times a \times b \)
Area = \( \frac{1}{2} \times 25 \times 25 \)
Area = \( \frac{1}{2} \times 625 \)
Area = 312.5 km2
Next, we calculate the population density:
Population density = \( \frac{Population}{Area} \)
Population density = \( \frac{100,000}{312.5} \)
Population density \approx 320 people/km2 (rounded to the nearest integer)