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Chapter 12: Problem 55

In solving a related rates problem, a key step is solving the derived equation for the unknown rate of change (once we have substituted the other values into the equation). Call the unknown rate of change \(X\). The derived equation is what kind of equation in \(X\) ?

Short Answer

Expert verified
The derived equation in a related rates problem can be of any type, such as linear, quadratic, higher-order polynomial, exponential, logarithmic, or trigonometric, depending on the relationships between the variables in the problem. The primary focus is on solving an equation involving the unknown rate of change, \(X\), and other known or given rates of change. The exact type of equation depends on the details of the problem.

Step by step solution

01

Identify the unknown rate of change

In a related rates problem, we generally have a given scenario where multiple variables are changing with respect to time (or another parameter), and we are asked to find the relationship between these rates of change. One of these rates will be the unknown rate of change, denoted by \(X\).
02

Understand the derived equation

The derived equation is an equation that results from taking the derivative of a quantity with respect to time (or another parameter) and using the chain rule to relate the rates of change of multiple variables in the problem. This equation will involve \(X\) as well as other known or given rates of change.
03

Identify the type of the derived equation

The derived equation obtained in a related rates problem can be of any type, depending on the relationships between the variables in the problem. It could be a linear, quadratic, or higher-order polynomial equation in terms of \(X\). Additionally, other types of equations such as exponential, logarithmic, or trigonometric equations could be formed, depending on the specifics of the problem. While there isn't a specific type of equation that will always be formed in a related rates problem, the primary focus is on solving an equation involving the unknown rate of change, \(X\), and other known or given rates of change. These equations will almost always be either an algebraic equation or transcendent equation involving \(X\), but the exact type depends on the details of the problem.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivative
When you're managing related rates problems, derivatives become your best friend. A derivative is a tool that helps us understand how a quantity changes as something else changes. Here, it shows the rate at which a variable changes with respect to another.
In essence, whenever you see something like \( \frac{dy}{dx} \) in math, that's a derivative. The 'dy' represents a small change in 'y', and 'dx' is a small change in 'x'. When we're talking about related rates, usually it's how something changes over time. So we might have \( \frac{dx}{dt} \) which represents the rate of change of 'x' with respect to time, 't'.
  • The derivative helps to express the dynamic relationship between variables.
  • It's critical because it provides a mathematical framework to understand "how fast" something is changing.

      • When working through related rates, recognizing how to derive an equation properly is essential. You'll often use derivatives to transform a static relationship between variables into a dynamic one, highlighting changes over time.
Rate of Change
Rate of change is a cornerstone in related rates problems. It gives a quantitative measure of how one quantity varies in relation to another.
Imagine a car traveling along the road; the rate of change of your car's position is what we commonly call speed. Similarly, in these math problems, we find the rate of change of one variable while other conditions or inputs continuously change.
  • Rate of change is expressed mathematically using derivatives.
  • Typical notation includes \( \frac{dy}{dt} \) for change over time.

Understanding the context of the problem is key. The physical scenario and its variables decide what kind of equations and rates you'll be dealing with. The focus is usually to solve for these unknown rates of change, integrating dynamics into your geometry or algebra.
Chain Rule
The chain rule is a pivotal concept when dealing with related rates. It helps calculate the derivative of a composition of functions - essentially functions within functions. When variables are related through such a composition, the chain rule finds the overall rate of change.
Picture a scenario where one variable influences another, and that secondary variable then affects a third variable. You need the chain rule to link these rates of change together.
  • The chain rule states that \( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \).
  • It's useful when the rate of change of a variable needs to pass through an intermediate variable.

The simplicity of the chain rule lies in multiplying derivatives, allowing the link between multiple changes to form naturally. In related rate problems, it helps relate different derivatives seamlessly, translating complex relationships into manageable equations.
Polynomial Equations
Polynomial equations frequently appear in related rate problems, often representing the dynamic relationships among changing variables.
Think of polynomial equations like \( ax^n + bx^{n-1} + \ldots + k = 0 \). They might seem complicated, but they can often represent the scenario at hand in straightforward terms, with terms built from one or more variables and constants.
  • Polynomial equations range from simple linear equations to more complex quadratics or higher degree polynomials.
  • In related rate problems, they form when the relationships involve non-linear changes among variables.

The beauty of polynomial equations comes from their ability to accommodate various changes and complexities in problems. Understanding how to manipulate and solve these equations helps unravel the related rates' mysteries, paving the way to find the elusive unknown rate of change.

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Most popular questions from this chapter

The following graph shows a striking relationship between the total prison population and the average combined SAT score in the United States: These data can be accurately modeled by $$S(n)=904+\frac{1,326}{(n-180)^{1.325}} . \quad(192 \leq n \leq 563)$$ Here, \(S(n)\) is the combined U.S. average SAT score at a time when the total U.S. prison population was \(n\) thousand. \(^{39}\) a. Are there any points of inflection on the graph of \(S ?\) b. What does the concavity of the graph of \(S\) tell you about prison populations and SAT scores?

A right circular conical vessel is being filled with green industrial waste at a rate of 100 cubic meters per second. How fast is the level rising after \(200 \pi\) cubic meters have been poured in? The cone has a height of \(50 \mathrm{~m}\) and a radius of \(30 \mathrm{~m}\) at its brim. (The volume of a cone of height \(h\) and crosssectional radius \(r\) at its brim is given by \(V=\frac{1}{3} \pi r^{2} h .\).)

Daily oil production by Pemex, Mexico's national oil company, can be approximated by \(q(t)=-0.022 t^{2}+0.2 t+2.9\) million barrels \(\quad(1 \leq t \leq 9)\) where \(t\) is time in years since the start of \(2000 .^{54}\) At the start of 2008 the price of oil was \(\$ 90\) per barrel and increasing at a rate of \(\$ 80\) per year. \(^{55}\) How fast was Pemex's oil (daily) revenue changing at that time?

The approximate value of subprime (normally classified as risky) mortgage debt outstanding in the United States can be approximated by $$A(t)=\frac{1,350}{1+4.2(1.7)^{-t}} \text { billion dollars } \quad(0 \leq t \leq 8)$$ \(t\) years after the start of \(2000^{53}\) Graph the derivative \(A^{\prime}(t)\) of \(A(t)\) using an extended domain of \(0 \leq t \leq 15 .\) Determine the approximate coordinates of the maximum and determine the behavior of \(A^{\prime}(t)\) at infinity. What do the answers tell you?

The following graph shows the approximate value of the United States Consumer Price Index (CPI) from December 2006 through July \(2007.31\) The approximating curve shown on the figure is given by $$I(t)=-0.04 t^{3}+0.4 t^{2}+0.1 t+202 \quad(0 \leq t \leq 7)$$ where \(t\) is time in months ( \(t=0\) represents December 2006). a. Use the model to estimate the monthly inflation rate in February \(2007(t=2) .\) [Recall that the inflation rate is \(\left.I^{\prime}(t) / I(t) .\right]\) b. Was inflation slowing or speeding up in February \(2007 ?\) c. When was inflation speeding up? When was inflation slowing? HINT [See Example 3.]
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