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.

Explanation:

Step1: Analyze the water - population relationship

We know that the total amount of water \(W\) is fixed, and the population is \(x\). The amount of water per person \(f(x)=\frac{W}{x}\) (in this case, \(W = 500\), so \(f(x)=\frac{500}{x}\)). As the population \(x\) increases (since the population is increasing over time), and the numerator (total water) is constant, we can use the properties of rational functions. For a function of the form \(y=\frac{k}{x}\) (\(k>0\)), as \(x\) increases, \(y\) decreases. So, as time passes and the population \(x\) increases, the amount of water per person \(f(x)\) will decrease.

Step2: Identify the function type

The function \(f(x)=\frac{500}{x}\) is of the form \(y = \frac{k}{x}\), where \(k = 500\) is a constant. This is the form of an inverse - variation (or reciprocal) function, which is a type of rational function. In the context of a unit on rational functions (implied by the hint), this function represents the relationship between population and water per person.

Answer:

1. As time passes and the population of the city increases, the availability of water per person will decrease. This is because the total amount of water is fixed, and when it is divided among a larger number of people, each person gets a smaller share. Mathematically, from the function \(f(x)=\frac{500}{x}\), as \(x\) (population) increases, \(f(x)\) (water per person) decreases.
2. Factors that could impact this include the rate of population growth (faster growth would lead to a quicker decrease in water per person), changes in water conservation efforts (if people conserve water, the effective total water available per person could be higher than predicted by just the population growth), and changes in the actual total water supply (e.g., if new water sources are found, the total water \(W\) would increase, and if there is a drought, \(W\) would decrease). The function \(f(x)=\frac{500}{x}\) is an inverse - variation (reciprocal) function, which is a type of rational function. This makes sense in the context of a unit on rational functions, as the relationship between two quantities where one is the reciprocal of the other is modeled by a rational function of this form.