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Algebra, STD1 EQ-Bank 13

A boat is purchased for $15 000. It depreciates in value by $3000 per year.

Let  \(V\) = Value of the boat in dollars, and  \(t\) = time in years.

  1. Complete the table of values below that models the relationship between the value of the boat and time in years.   (1 mark)
      
    \(\begin{array}{|c|c|c|c|c|c|c|} \hline \vphantom{\dfrac{1}{1}}\quad t \quad & \quad 0 \quad & \quad 1 \quad & \quad 2 \quad & \quad 3 \quad & \quad 4 \quad & \quad 5 \quad \\[6pt] \hline \vphantom{\dfrac{1}{1}}V & 15\,000 &  & 9000 &  & 3000 &  \\[12pt] \hline \end{array}\)

  2. Using the table of values from (a), neatly graph the value of the boat from 0 to 5 years on the grid below.   (2 marks)

      

     

  3. Identify ONE limitation of this linear model.   (1 mark)

Show Answers Only

a.    \(\text{Table of values}\)

\begin{array}{|c|c|c|c|c|c|c|} \hline t & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline V & 15\ 000 & \textbf{12 000} & 9000 & \textbf{6000} & 3000 & \ \ \ \ \textbf{0}\ \ \ \  \\ \hline \end{array}

b.  

     

c.     \(\text{Limitations could include ONE of the following:}\)

    • \(\text{The model predicts the boat has zero value after 5 years, which is}\)
      \(\text{unrealistic as most boats retain some value.}\)
    • \(\text{Beyond 5 years the model would predict a negative value, which is}\)
      \(\text{not possible.}\)
    • \(\text{The model assumes a constant rate of depreciation, but in reality}\)
      \(\text{a boat may depreciate more quickly in early years.}\)
Show Worked Solution

a.    \(\text{Table of values}\)

\begin{array}{|c|c|c|c|c|c|c|} \hline t & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline V & 15\ 000 & \textbf{12 000} & 9000 & \textbf{6000} & 3000 & \ \ \ \ \textbf{0}\ \ \ \  \\ \hline \end{array}

 
b.  

     

c.     \(\text{Limitations could include ONE of the following:}\)

    • \(\text{The model predicts the boat has zero value after 5 years, which is}\)
      \(\text{unrealistic as most boats retain some value.}\)
    • \(\text{Beyond 5 years the model would predict a negative value, which is}\)
      \(\text{not possible.}\)
    • \(\text{The model assumes a constant rate of depreciation, but in reality}\)
      \(\text{a boat may depreciate more quickly in early years.}\)

Algebra, STD1 EQ-Bank 20

Sunny SUPs Pty Ltd charges a $2.50 online booking fee plus $25 per hour for stand-up paddle board hire.

A student claims that Sunny SUPs' income can be modelled using a linear equation.

Is the student correct? Justify your answer.   (2 marks)

Show Answers Only

\(\text{Yes, the student is correct.}\)

\(\text{Correct answers could include any ONE of the following justifications:}\)

    • \(\text{The income increases by a constant amount of \$25 for each additional}\)
      \(\text{hour hired.}\)
    • \(\text{The income can be written in the form }\  I=25h+2.50,\text{which is a linear}\)
      \(\text{equation.}\)
    • \(\text{The graph of income against hours hired would be a straight line.}\)
    • \(\text{The rate of change of income is constant at \$25 per hour.}\)
Show Worked Solution

\(\text{Yes, the student is correct.}\)

\(\text{Correct answers could include any ONE of the following justifications:}\)

    • \(\text{The income increases by a constant amount of \$25 for each additional}\)
      \(\text{hour hired.}\)
    • \(\text{The income can be written in the form }\  I=25h+2.50,\text{which is a linear}\)
      \(\text{equation.}\)
    • \(\text{The graph of income against hours hired would be a straight line.}\)
    • \(\text{The rate of change of income is constant at \$25 per hour.}\)

Algebra, STD1 EQ-Bank 11

A farmer in Western Australia observes that the rodent population on his property is doubling every two weeks.

A student claims that a linear model is NOT appropriate to predict the rodent population over time.

Is the student correct? Justify your answer.   (2 marks)

Show Answers Only

\(\text{Yes, the student is correct.}\)

\(\text{Correct justification could include ONE of the following:}\)

    • \(\text{The rodent population is doubling every two weeks, which means it}\)
      \(\text{is increasing by a constant multiplier, not a constant amount — this}\)
      \(\text{is exponential growth, not linear.}\)
    • \(\text{A linear model increases by the same amount each time period, but}\)
      \(\text{the rodent population increases by a larger amount each fortnight as}\)
      \(\text{the population grows.}\)
    • \(\text{The graph of the rodent population over time would be an upsloping}\)
      \(\text{exponential curve, not a straight line.}\)
    • \(\text{A linear model would significantly underestimate the rodent population}\)
      \(\text{over time.}\)
Show Worked Solution

\(\text{Yes, the student is correct.}\)

\(\text{Correct justification could include ONE of the following:}\)

    • \(\text{The rodent population is doubling every two weeks, which means it}\)
      \(\text{is increasing by a constant multiplier, not a constant amount — this}\)
      \(\text{is exponential growth, not linear.}\)
    • \(\text{A linear model increases by the same amount each time period, but}\)
      \(\text{the rodent population increases by a larger amount each fortnight as}\)
      \(\text{the population grows.}\)
    • \(\text{The graph of the rodent population over time would be an upsloping}\)
      \(\text{exponential curve, not a straight line.}\)
    • \(\text{A linear model would significantly underestimate the rodent population}\)
      \(\text{over time.}\)