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You are working part-time for an electronics company while going to high school. The following table shows the hourly wage, [tex]\( w(t) \)[/tex], in dollars, that you earn as a function of time, [tex]\( t \)[/tex]. Time is measured in years since the beginning of 2004 when you started working.

\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
\begin{tabular}{l}
Time, [tex]\( t \)[/tex] (Years since 2004)
\end{tabular} & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline
Hourly wage, [tex]\( w(t) \)[/tex] (\$) & 9.00 & 9.90 & 10.90 & 12.00 & 13.20 & 14.50 \\
\hline
\end{tabular}

What is the growth factor?

A. 0.90
B. 1
C. 1.2
D. 1.1

Please select the best answer from the choices provided.



Answer :

GinnyAnswer
To determine the growth factor of your hourly wage over time, we need to analyze the changes in your wages across different years and calculate the average growth factor.

Firstly, let's list out the given data points, which denote your hourly wage for specific years:
- Time [tex]\( t = 0 \)[/tex]: Wage [tex]\( w = \$9.00 \)[/tex]
- Time [tex]\( t = 1 \)[/tex]: Wage [tex]\( w = \$9.90 \)[/tex]
- Time [tex]\( t = 3 \)[/tex]: Wage [tex]\( w = \$10.90 \)[/tex]
- Time [tex]\( t = 4 \)[/tex]: Wage [tex]\( w = \$12.00 \)[/tex]
- Time [tex]\( t = 5 \)[/tex]: Wage [tex]\( w = \$13.20 \)[/tex]
- Time [tex]\( t \)[/tex] (not listed in the table above but given): Wage [tex]\( w = \$14.50 \)[/tex]

To compute the growth factors between each successive wage increment, we follow these steps:

1. Calculate the growth factor from [tex]\( t = 0 \)[/tex] to [tex]\( t = 1 \)[/tex]:
[tex]\[ \text{Growth Factor}_{0 \to 1} = \frac{w_{1}}{w_{0}} = \frac{9.90}{9.00} \approx 1.1 \][/tex]

2. Calculate the growth factor from [tex]\( t = 1 \)[/tex] to [tex]\( t = 3 \)[/tex]:
[tex]\[ \text{Growth Factor}_{1 \to 3} = \frac{w_{3}}{w_{1}} = \frac{10.90}{9.90} \approx 1.101 \][/tex]

3. Calculate the growth factor from [tex]\( t = 3 \)[/tex] to [tex]\( t = 4 \)[/tex]:
[tex]\[ \text{Growth Factor}_{3 \to 4} = \frac{w_{4}}{w_{3}} = \frac{12.00}{10.90} \approx 1.1009 \][/tex]

4. Calculate the growth factor from [tex]\( t = 4 \)[/tex] to [tex]\( t = 5 \)[/tex]:
[tex]\[ \text{Growth Factor}_{4 \to 5} = \frac{13.20}{12.00} = 1.1 \][/tex]

5. Calculate the growth factor from [tex]\( t = 5 \)[/tex] onward:
[tex]\[ \text{Growth Factor}_{5 \to (next)} = \frac{14.50}{13.20} \approx 1.0985 \][/tex]

To find the representative growth factor, we need to average these calculated growth factors:
[tex]\[ \text{Average Growth Factor} = \frac{(1.1 + 1.101 + 1.1009 + 1.1 + 1.0985)}{5} \approx 1.10008 \][/tex]

The closest value from the given choices that match this average growth factor is [tex]\( 1.1 \)[/tex]. Hence, the best answer is:

d. 1.1

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