Question

summative grade: after watching the video, \life in a bubble,\ answer the following questions in complete sentences this assignment is due friday 12/5 at 11:59 pm imagine a city with a fixed amount of water that must be shared among an increasing population. 1. how might the availability of water per person change over time? 2. what factors could impact this? what type of function might represent this relationship (hint: what is the name of this unit)? use the graph $f(x) = \frac{500}{x}$, where x is the population number, and $f(x)$ is the amount of available water. graph of $f(x) = \frac{500}{x}$ with x (population) on the x - axis and f(x) (amount of available water) on the y - axis 1. as the population gets larger and larger, what happens to the amount of water available? if the population doubles trrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr 2. what happens to the amount of available water when the population doubles (is multiplied by 2)?

Explanation:

Question 1 (As the population gets larger...):

Step1: Analyze the function

The function is \( f(x)=\frac{500}{x} \), where \( x \) is population and \( f(x) \) is water per person. This is a rational (inverse variation) function.

Step2: Observe as \( x \) increases

As \( x \) (population) approaches \( +\infty \), we evaluate the limit: \( \lim_{x
ightarrow\infty}\frac{500}{x} \). Since the numerator is constant and the denominator grows without bound, the value of \( \frac{500}{x} \) approaches 0.

Step1: Let original population be \( x \), new population be \( 2x \)

Original water per person: \( f(x)=\frac{500}{x} \). New water per person: \( f(2x)=\frac{500}{2x} \).

Step2: Simplify \( f(2x) \)

Simplify \( \frac{500}{2x} \) to \( \frac{1}{2}\times\frac{500}{x} \), which is \( \frac{1}{2}f(x) \).

Answer:

As the population gets larger and larger, the amount of available water per person approaches 0 (it gets smaller and smaller, getting closer to 0 but never actually reaching 0 in a practical sense for a finite population, but mathematically approaches 0 as population becomes infinitely large).

Question 2 (When population doubles...):