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Matias models the relationship between Australia's gross domestic product per capita

\[P\] (in dollars) and time
\[T\] (in years after 1950) as
\[P=1800\cdot 1.067^T\].
He wanted to create a graph to show the relationship until the year 2000. Here is his work:
An exponential function increases as x increases on an x y coordinate plane. The x axis scales by five from zero to fifty. The y axis scales by five thousand from zero to forty five thousand. The x axis is labeled Years since 1950. The graph starts at the y axis at (zero, one thousand eight hundred) passing through the points around (fifteen, five thousand) and (fifty, forty five thousand).
\[\small{5}\]
\[\small{10}\]
\[\small{15}\]
\[\small{20}\]
\[\small{25}\]
\[\small{30}\]
\[\small{35}\]
\[\small{40}\]
\[\small{45}\]
\[\small{5000}\]
\[\small{10{,}000}\]
\[\small{15{,}000}\]
\[\small{20{,}000}\]
\[\small{25{,}000}\]
\[\small{30{,}000}\]
\[\small{35{,}000}\]
\[\small{40{,}000}\]
What mistakes did Matias make when drawing the graph?
Choose all answers that apply:
Choose all answers that apply:
(Choice A) Inappropriate scale for
\[x\]-axis
A
Inappropriate scale for
\[x\]-axis
(Choice B) Inappropriate scale for
\[y\]-axis
B
Inappropriate scale for
\[y\]-axis
(Choice C) Wrong/missing label for
\[x\]-axis
C
Wrong/missing label for
\[x\]-axis
(Choice D) Wrong/missing label for
\[y\]-axis
D
Wrong/missing label for
\[y\]-axis
(Choice E) Matias didn't make any mistake
E
Matias didn't make any mistake

1 answer

To analyze Matias's graph, we need to consider the function he presented:

\[ P = 1800 \cdot 1.067^T \]

Where:

  • \( P \) is Australia's gross domestic product per capita in dollars.
  • \( T \) is the number of years after 1950.

  1. Understanding \( T \):

    • For the year 2000, \( T = 2000 - 1950 = 50 \).
    • This means Matias should plot values for \( T \) ranging from \( 0 \) (1950) to \( 50 \) (2000).

  2. Calculating \( P \):

    • At \( T = 0 \): \[ P = 1800 \cdot 1.067^0 = 1800 \]
    • At \( T = 15 \): \[ P = 1800 \cdot (1.067)^{15} \approx 1800 \cdot 2.709 = 4880.2 \] (close to 5000, but not exact)
    • At \( T = 50 \): \[ P = 1800 \cdot (1.067)^{50} \approx 1800 \cdot 11.663 = 21000 \] (not 45000)

  3. Evaluating Scale Choices:

    • X-axis scale isn’t inappropriately scaled for the range of \( T \) from 0 to 50, which is a standard range in this context.
    • Y-axis scale should reflect the actual GDP per capita values. Given the calculated numbers:
      • It starts at 1800 and the maximum at \( T = 50 \) is around 21000. Therefore, the scale labeled up to 45000 is inappropriate.

  4. Labels:

    • The x-axis is labeled as "Years since 1950," which is correct.
    • The y-axis should indicate GDP per capita in dollars, which is likely missing or incorrect as described.

Based on this analysis, we can identify the potential mistakes made by Matias:

  • (Choice B) Inappropriate scale for \( y \)-axis
  • (Choice D) Wrong/missing label for \( y \)-axis

Therefore, the selected choices of mistakes by Matias are B and D.