AOS4 Topic 2: Rates and Rational Functions

Rates

Rates are the measurement of changes in one thing in relation to one. They are typically expressed in terms of a ratio of two distinct numbers, for example speed (distance traveled per minute) or the change of an amount over time. Rates are crucial in a variety of areas, such as Physics, mathematics and economics.

Speed/Velocity

Formula: \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \)

Explanation: The rate at which an object changes location relative to time is measured as speed or velocity. It is stated in terms of distance traveled per unit of time. For example, if a car drives 100 kilometers in two hours, its speed is \( \frac{100 \, km}{2 \, hours} = 50 \, km/h \).

Acceleration

Formula: \( \text{Acceleration} = \frac{\text{Change in Velocity}}{\text{Time}} \)

Explanation: Acceleration is the term used to describe the speed at which an object's velocity alters over the course of time. It is the rate that velocity changes. Acceleration may either be negative (moving faster) or positive (moving faster), negative (moving slower) or even zero. For example If a car speed between 20 and 30 m/s within 5 seconds, its acceleration will be \( \frac{(30 \, m/s - 20 \, m/s)}{5 \, s} = 2 \, m/s^2 \).

Growth Rate

Formula: \( \text{Growth Rate} = \left( \frac{\text{Final Value} - \text{Initial Value}}{\text{Initial Value}} \right) \times 100\% \)

Explanation: Growth rate refers to the rate at which a quantity rises over time. It is often used in economics, finance, and demographic studies to examine variations in values like GDP, investment returns, and population numbers. For example, if a city's population grows from 1,000 to 1,200 in one year, the growth rate will be \( \left( \frac{1,200 - 1,000}{1,000} \right) \times 100\% = 20\% \).

Interest Rate

Formula: \( \text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \)

Explanation: Interest rate represents the cost of borrowing money or the return on investment. It is typically expressed as a percentage per unit of time (usually per year). The formula calculates the amount of interest earned or paid based on the principal amount, the interest rate, and the time period. For example, if you borrow $1,000 at an annual interest rate of 5% for one year, the interest would be $1,000 \times 0.05 \times 1 = $50.

Population Growth Rate

Formula: \[ \text{Population Growth Rate} = \left( \frac{\text{Final Population} - \text{Initial Population}}{\text{Initial Population}} \right) \times 100\% \]

Explanation: Population growth rate measures the rate at which a population increases or decreases over time. It is calculated as the percentage change in population size relative to the initial population size. For example, if a town's population grows from 10,000 to 12,000 in one year, the population growth rate would be \(\left( \frac{12,000 - 10,000}{10,000} \right) \times 100\% = 20\%\).

Inflation Rate

Formula: \[ \text{Inflation Rate} = \left( \frac{\text{Current Price Index} - \text{Previous Price Index}}{\text{Previous Price Index}} \right) \times 100\% \]

Explanation: Inflation rate measures the rate at which the general level of prices for goods and services is rising, resulting in a decrease in purchasing power over time. It is typically expressed as a percentage change in a price index over time. For example, if the price index increases from 150 to 160 over one year, the inflation rate would be \(\left( \frac{160 - 150}{150} \right) \times 100\% = 6.67\%\).

Unemployment Rate

Formula: \[ \text{Unemployment Rate} = \left( \frac{\text{Number of Unemployed}}{\text{Labor Force}} \right) \times 100\% \]

Explanation: Unemployment rate measures the percentage of the labor force that is unemployed and actively seeking employment. It is an important indicator of the health of an economy. For example, if a country has 5 million unemployed individuals out of a labor force of 100 million, the unemployment rate would be \(\left( \frac{5 \text{ million}}{100 \text{ million}} \right) \times 100\% = 5\%\).

Reaction Rate

Formula: \[ \text{Reaction Rate} = \frac{\text{Change in Concentration}}{\text{Time}} \]

Explanation: In chemistry, reaction rate refers to the speed at which reactants are consumed or products are formed in a chemical reaction. It is typically measured as the change in concentration of a reactant or product per unit time. For example, if the concentration of a reactant decreases from 0.2 M to 0.1 M in 10 seconds, the reaction rate would be \(\frac{0.1 \, \text{M} - 0.2 \, \text{M}}{10 \, \text{s}} = -0.01 \, \text{M/s}\).

Rate of Gas Effusion

Formula: \[ \text{Rate of Effusion} = \frac{\text{Volume of Gas}}{\text{Time}} \]

Explanation: The rate of gas effusion refers to the rate at which a gas escapes through a small hole into a vacuum. It is influenced by factors such as the molar mass of the gas and the size of the hole. According to Graham's law of effusion, lighter gases effuse faster than heavier gases. For example, if 1 liter of a gas effuses through a hole in 2 minutes, its effusion rate would be \(\frac{1 \, \text{L}}{2 \, \text{min}} = 0.5 \, \text{L/min}\).

Rates and Rational Functions

Rates

Rates refer to the measurement of change in one quantity with respect to another quantity. They are often expressed as a ratio of two different quantities, such as distance traveled per unit of time (speed), or change in a quantity over time (growth rate, interest rate, etc.). Rates can be represented mathematically as derivatives in calculus or as fractions, where the numerator represents the change in the quantity being measured and the denominator represents the corresponding change in time or another independent variable.

Rational Functions

Rational functions are functions that can be expressed as the quotient of two polynomial functions. In other words, a rational function is a ratio of two polynomial expressions. They can be represented in the form \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) \) is not the zero polynomial. Rational functions often arise in various mathematical models, including those involving rates of change.

Relationship between Rates and Rational Functions

Here are some examples where rates and rational functions are related:

  1. Motion Problems: In physics, the motion of an object often involves rates of change, such as velocity and acceleration. These rates can be represented using rational functions. For instance, the velocity of an object in free fall might be represented by a rational function involving time.
  2. Population Growth Models: Models of population growth often involve rates of change, such as birth rates and death rates. These rates can be used to construct rational functions that describe how the population changes over time.
  3. Finance and Economics: In finance and economics, rates such as interest rates, inflation rates, and growth rates are crucial for understanding various phenomena. These rates can be modeled using rational functions to analyze their impact on economic variables over time.

Relationship between Rates and Rational Functions

Rates and rational functions are interconnected concepts that often appear together, particularly in problems involving rates of change over time.

Rates

Rates represent the measurement of change in one quantity with respect to another quantity. They are typically expressed as ratios and can be represented mathematically as derivatives in calculus.

Example: Speed

The speed of an object is a classic example of a rate. It is defined as the distance traveled per unit of time and can be represented by the following formula: \[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]

Rational Functions

Rational functions are functions that can be expressed as the quotient of two polynomial functions. They often arise in problems involving rates of change, as they can describe how one quantity changes relative to another.

Example: Population Growth

Consider a population that grows at a rate proportional to its size. The population growth can be modeled by a rational function: \[ P(t) = \frac{P_0}{1 - rP_0t} \] where:

  • \( P(t) \): Population at time \( t \)
  • \( P_0 \): Initial population
  • \( r \): Growth rate
  • \( t \): Time

Relationship

The relationship between rates and rational functions is evident in their mathematical representation. Rates often involve the division of two quantities (e.g., distance and time), leading to rational expressions. Moreover, rational functions can describe how quantities change over time, which aligns with the concept of rates of change.

Graphical Representation

Let's visualize the relationship with a graphical example. Consider the rational function: \[ f(x) = \frac{3x-5}{-2x+2} \]

The graph of this rational function demonstrates how the rate of change of the function (slope of the curve) varies with the independent variable \(x\).

Example 1

A train travels 300 miles in 3 hours. What will be the speed of train in miles per second?

Solution

Given that: \(distance= 300 miles, time = 3 hours\)

\(time = 3*3600 = 10800 seconds\) Speed = \( \frac{300 \, miles}{ 10800\, seconds} = 0.0278 \, miles/s \)

Example 2

Calculate the distance if A train travels from City A to City B, a distance of 800 kilometers, in 5 hours.

Given: Distance = 800 kilometers, Time = 5 hours

Solution: Speed = \( \frac{800 \, \text{km}}{5 \, \text{hours}} = 160 \, \text{km/h} \)

Example 3

As Steven’s ice block melts, it forms a circular puddle on the floor. The radius of the puddle increases at a rate of 3 cm/min. When its radius is 2 cm, find the rate at which the area of the puddle is increasing.

Solution:

The area, \( A \), of a circle is given by \( A = \pi r^2 \), where \( r \) is the radius of the circle.

The rate of increase of the radius is \( \frac{dr}{dt} = 3 \) cm/min.

Using the chain rule, the rate of increase of the area is: \[ \frac{dA}{dt} = \frac{dA}{dr} \cdot \frac{dr}{dt} = 2\pi r \times 3 = 6\pi r \]

When \( r = 2 \), \[ \frac{dA}{dt} = 12\pi \]

Hence the area of the puddle is increasing at \( 12\pi \) cm\(^2\)/min.

Example 4

A rectangular prism is being filled with water at a rate of \(0.00042 m^3/s\). Find the rate at which the height of the water is increasing.

Solution

Let t be the time in seconds after the prism begins to fill. Let V m3 be the volume of water at time t, and let h m be the height of the water at time t.

We are given that dV/dt = 0.00042 and V = 6h.

Using the chain rule, the rate at which the height is increasing is:

dh/dt = (dh/dV) × (dV/dt)

Since V = 6h, we have dV/dh = 6 and so dh/dV = 1/6.

dh/dt = (1/6) × 0.00042 = 0.00007 m/s

i.e. the height is increasing at a rate of 0.00007 m/s.

Exercise &&1&& (&&1&& Question)

A runner completes a 5-kilometer race in 25 minutes. What will be his speed?

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Exercise &&2&& (&&1&& Question)

A metal cube is being heated so that the side length is increasing at the rate of 0.02 cm per hour. Calculate the rate at which the volume is increasing when the side length is 5 cm.

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Exercise &&3&& (&&1&& Question)

What is the formula of distance?

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