Question

The amount, \(A\), of pollution in a certain city is a function of the population \(P\), with \(A=100 P^{0.3}\). The population is growing over time, and \(P=10000+2000 t\), with \(t\) in years since 2000 . Express the amount of pollution \(A\) as a function of time \(t\).

Short answer

Question: Given the pollution function \(A(P) = 100 P^{0.3}\), where A is the amount of pollution and P is the population, and the population function \(P(t) = 10000 + 2000t\), where P is the population and t is the time, find the function for the amount of pollution A as a function of time t. Answer: The function for the amount of pollution A as a function of time t is \(A(t) = 100(10000 + 2000t)^{0.3}\).

Step-by-step solution

Identify the given functions

We are given the following functions: \(A(P) = 100 P^{0.3}\), which represents the amount of pollution for a given population P, and \(P(t) = 10000 + 2000t\), which represents the population for a given time t.

Substitute the population function into the pollution function

In order to express the amount of pollution \(A\) as a function of time \(t\), we need to substitute the population function \(P(t)\) into the pollution function \(A(P)\): \(A(t) = A(P(t)) = 100(10000 + 2000t)^{0.3}\)

Simplify the function

Now, we simplify the function: \(A(t) = 100(10000 + 2000t)^{0.3}\) This is the function for the amount of pollution \(A\) as a function of time \(t\).

Key concepts

Exponential Growth

When we talk about exponential growth, we are usually referring to a pattern where a quantity increases over time at a rate proportional to its current value. In simpler terms, the bigger the quantity gets, the faster it grows. This is common in situations like population, investments, and yes—even pollution levels! For example, in our exercise, you can observe that the pollution level, which depends on population, increases as the population itself grows over time.

Imagine doubling your pocket money each day; today it's $1, tomorrow it'll be $2, followed by $4 the next day—you get the drift. Exponential growth can get big really fast. In mathematical terms, this is often expressed using powers or exponents. For example, in the pollution function, we have an exponent of 0.3, showing how fast the pollution increases relative to the population size.

Understanding exponential growth is crucial because it helps us prepare for situations where growth might be unsustainable at some point. It sets the stage for strategies to manage such situations effectively.

Function Composition

Function composition is just like building with Lego blocks. You take one function or rule and stick it onto another to create a new action or process. In algebra, we represent functions by f(x) or g(x), and when we compose them, it looks like f(g(x)). This means we're inputting g(x) into f.

Imagine a machine that paints toys (function f) and another that assembles toys (function g). If 'g' assembles and then 'f' paints, you're composing functions to get lovely painted toys! Similarly, for the exercise above, to express pollution over time, we take the population as a function of time, P(t), and use it as the input in the pollution function, A(P). Thus, A(P(t)) gives us pollution over time.

Function composition is important because it helps solve complex problems by breaking them down into simpler steps. Here, by first understanding how the population grows and then how pollution depends on the population, we can effectively model pollution over time.

Population Models

Population models are mathematical representations, often using functions, to predict how a population changes over time. They've become a crucial part of planning in fields like ecology, urban planning, and resource management.

In our exercise, we have a linear population model represented by the equation P(t) = 10000 + 2000t. Linear models are often used when you expect consistent growth over time. If the population grows by the same amount each year, a linear model is both simple and effective. However, in real-life scenarios, populations might grow more sporadically, potentially switching to exponential models.

Using these models helps organizations anticipate needs for resources, control pollution, and optimize the allocation of services. By employing mathematical models for such forecasts, we strive for a balance between growth and sustainability in our communities.