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Quantitative Skills and Reasoning Activity Workbook

Megan Patnott

Section 12.2 Introduction to Power Models

Activity 12.2.1.

Recall from Section 12.1 that a power model is a relationship between two quantities that can be written in the form
\begin{equation*} Q_2 = c \times Q_1^p\text{,} \end{equation*}
where \(Q_1\) and \(Q_2\) are the two quantities, \(c\) is a number called the coefficient and \(p\) is a number called the power. The quantitites \(Q_1\) and \(Q_2\) are variables, and the coefficient and the power are numbers.
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Determine which of the following are power models and which are not. Briefly explain your answers in words.
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Activity 12.2.2.

The volume of a cube can be modeled by the formula
\begin{equation*} V = s^3 \end{equation*}
where \(s\) is the length of a side of the cube and \(V\) is the cube’s volume.
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(a)

Explain why the volume of a cube is a power model, and identify its quantities, coefficient, and power.
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(b)

Sketch a graph showing the relationship between the volume of a cube and its side lengths.
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(c)

Confirm that the volume of a cube is not a linear model using average rates of change.
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(d)

Confirm that the volume of a cube is not an exponential model using percent change or change factors.
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(e)

In order to find the side length of a cube with volume \(8 \text{m}^3\text{,}\) we can use one of two equivalent strategies:
  1. Find the cube root of 8, \(\sqrt[3]{8}\text{.}\)
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  2. Find the \(\frac{1}{3}\) power of 8, \(8^{1/3}\text{.}\)
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Find the side length of a cube with volume \(10 \text{ ft}^3\text{.}\)
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Power models with positive powers are often expressed in words using the language of proportionality, or by saying that one quantity varies directly with another.
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Example 12.2.1.

If the energy, \(E\text{,}\) expended by a swimming dolphin varies directly with the cube of its speed, \(v\text{,}\) then
\begin{equation*} E = cv^3\text{,} \end{equation*}
with \(c\) an unknown number, is a power model for \(E\text{.}\)
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Example 12.2.2.

The braking distance, \(d\text{,}\) of an Alfa Romeo on dry pavement is proportional to the square of its speed (\(v\)). An Alfa Romeo going 70 mph require 177 feet to stop. What is a power model for \(d\text{?}\)
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Solution: The first sentence gives us that the power model has the structure:
\begin{equation*} d = cv^2\text{,} \end{equation*}
where \(c\) is an unknown number. The second sentences gives us the information we need to find \(c\text{.}\) We plug in \(v = 70\) and \(d = 177\) to get
\begin{equation*} 177 = c(70)^2\text{.} \end{equation*}
Solving for \(c\) gives us that \(c \approx 0.036\text{.}\) Now we can give the final power model: \(d = 0.036v^2\text{.}\)
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Activity 12.2.3.

In each of the following situations, set up the power model being described, including using the information given to find the coefficient. Then, use your power model to answer the question.
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(a)

\(C\) is proportional to the square of \(d\text{.}\) When \(d = 3\text{,}\) \(C = 135\text{.}\) What is \(C\) when \(d = 7\text{?}\)
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(b)

\(T\) varies directly with the cube power of \(u\text{.}\) When \(u = 2\text{,}\) \(T = 56\text{.}\) What is \(u\) when \(T = 875\text{?}\)
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(c)

The electrical current, in amperes, in a circuit varies directly as the voltage. When 27 volts are applied, the current is 9 amperes. What is the current when 39 volts are applied?
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Activity 12.2.4.

When an aircraft takes off, it accelerates until it reaches its takeoff speed \(V\text{.}\) In doing so it uses up a distance \(R\) of the runway, where \(R\) is proportional to the square of the takeoff speed. An aircraft with a takeoff speed of 100 mph needs 1639 feet of runway.
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(a)

Find a formula for a power model for the relationship between \(R\) and \(V\text{.}\)
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(b)

Suppose that typical takeoff speeds for aircraft are between 60 and 190 mph. Sketch a graph showing the relationship between the takeoff speed and runway length needed for takeoff for aircraft.
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(e)

For this problem, use your power model formula, but ignore the aircraft context. Find all solutions to the equation you get when \(R = 1000\text{.}\)
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Activity 12.2.5.

The graphs of power models with positive integer powers are very predictable! Let’s find the pattern.
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(a)

Use Desmos to graph each of the power models listed below. What patterns do you notice?
\begin{equation*} y = x^2 \qquad y = x^3 \qquad y = x^4 \qquad y = x^5 \qquad y = x^6 \end{equation*}
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(b)

Use Desmos to graph each of the power models listed below. What patterns do you notice?
\begin{equation*} y = -x^2 \qquad y = -x^3 \qquad y = -x^4 \qquad y = -x^5 \qquad y = -x^6 \end{equation*}
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(c)

Use the patterns you noticed in the two parts above and what you already know about linear models to fill in each cell of the table with a sketch of the graph.
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Table 12.2.3. Shapes of Power Model Graphs
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Positive Coefficient Negative Coefficient
\(y = mx\)  
 
 
 
 		  
 
 
 
 
\(y = ax^{\text{even power}}\)  
 
 
 
 		  
 
 
 
 
\(y = ax^{\text{odd power}}\)  
 
 
 
 		  
 
 
 
 
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(d)

Explain why equations involving a power model with an odd power only have one solution.
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(e)

Suppose that we want to find all solutions to the equation
\begin{equation*} -96 = -6x^4\text{.} \end{equation*}
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(i)
Use a sketch of the graph of \(y = -6x^4\) to explain why there are two solutions. Then, use Desmos to find both solutions.
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(ii)
Find both solutions with algebra.
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(iii)
Camila consistently gets problems like this one wrong. Write a set of steps for Camila to follow that will help her find all solutions to equations involving a power model with an even power using algebra.
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