Problem 92 - Elementary Algebra

Question

In \(2000,30,530,000\) tons of yard waste went into the municipal waste stream. In 2010 , \(33,400,000\) tons of yard waste went into the municipal waste stream. Find the average rate of change of the amount of yard waste going into the municipal waste stream per year.

Step-by-Step Solution

Answer

287,000 tons per year

Step 1: Identify the given data

The problem gives the amount of yard waste that went into the municipal waste stream in 2000 and 2010. Let the amount in 2000 be denoted as \[{y_0} = 30,530,000\]. Let the amount in 2010 be denoted as \[{y_1} = 33,400,000\]. Let the two corresponding years be \[{x_0} = 2000\] and \[{x_1} = 2010\].

Step 2: Calculate the change in yard waste

Find the difference in the amount of yard waste between 2010 and 2000. \[\Delta y = y_1 - y_0 = 33,400,000 - 30,530,000 = 2,870,000\]

Step 3: Calculate the change in time

Find the difference in years between 2010 and 2000. \[\Delta x = x_1 - x_0 = 2010 - 2000 = 10\]

Step 4: Calculate the average rate of change

The average rate of change is the change in yard waste divided by the change in time. \[\text{Average rate of change} = \frac{\Delta y}{\Delta x} = \frac{2,870,000}{10} = 287,000\text{ tons per year}\]

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In this problem, we use algebraic concepts to find the average rate of change.
Here, we deal with variables representing yard waste and time.
This involves basic arithmetic operations like subtraction and division.
Understanding algebra is crucial as it simplifies how we express mathematical relationships and solve equations.

Municipal Waste

Municipal waste refers to the waste products that are collected and managed by a city or town, commonly known as garbage or trash.
It includes various types of waste like household waste, yard waste, food waste, and general rubbish.
In this exercise, the focus is on yard waste, which is a significant component of municipal waste because it is produced in large quantities, especially in areas with plentiful green spaces.

Rate of Change

The rate of change is a fundamental concept that describes how one quantity changes in relation to another.
It is calculated by dividing the change in one variable by the change in another.
In this exercise, we find the average rate of change of yard waste that enters the municipal waste stream over a decade.
The formula used is: \(\text{Average rate of change} = \frac{\text{Change in yard waste}}{\text{Change in time}}\). This helps us understand the trend or average rate at which the yard waste amount increased per year from 2000 to 2010.

Yard Waste

Yard waste includes organic waste from gardens and yards such as leaves, grass clippings, tree branches, and other greenery.
Proper management of yard waste is essential as it constitutes a large portion of municipal waste streams.
It can be recycled or composted, reducing the amount sent to landfills.
By tracking the rate of change in yard waste, municipalities can improve their waste management strategies and encourage recycling and composting to minimize landfill use.