Question
Benjamin invests
\[\$400\] in a savings account that earns
\[5\%\] interest each year. He models the relationship between the time
\[T\] since he opened the account (in years) and the account's balance
\[B\] (in dollars) as
\[B=400\cdot1.05^T\].
He wanted to create a graph to show the account's balance over the first
\[10\] years. Here is his work:
A line increasing as x increases on an x y coordinate plane. The x axis scales by one from zero to eight. The y axis scales by one hundred from negatve six hundred to six hundred. The y axis is labeled Balance in dollars. The x axis is labeled time in years. The line starts at the y axis at (zero, four hundred) passing through the point (four and one half, five hundred) before reaching around (eight, six hundred).
\[\small{1}\]
\[\small{2}\]
\[\small{3}\]
\[\small{4}\]
\[\small{5}\]
\[\small{6}\]
\[\small{7}\]
\[\small{100}\]
\[\small{200}\]
\[\small{300}\]
\[\small{400}\]
\[\small{500}\]
\[\small{\llap{-}200}\]
\[\small{\llap{-}300}\]
\[\small{\llap{-}400}\]
\[\small{\llap{-}500}\]
What mistakes did Benjamin 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
\[\$400\] in a savings account that earns
\[5\%\] interest each year. He models the relationship between the time
\[T\] since he opened the account (in years) and the account's balance
\[B\] (in dollars) as
\[B=400\cdot1.05^T\].
He wanted to create a graph to show the account's balance over the first
\[10\] years. Here is his work:
A line increasing as x increases on an x y coordinate plane. The x axis scales by one from zero to eight. The y axis scales by one hundred from negatve six hundred to six hundred. The y axis is labeled Balance in dollars. The x axis is labeled time in years. The line starts at the y axis at (zero, four hundred) passing through the point (four and one half, five hundred) before reaching around (eight, six hundred).
\[\small{1}\]
\[\small{2}\]
\[\small{3}\]
\[\small{4}\]
\[\small{5}\]
\[\small{6}\]
\[\small{7}\]
\[\small{100}\]
\[\small{200}\]
\[\small{300}\]
\[\small{400}\]
\[\small{500}\]
\[\small{\llap{-}200}\]
\[\small{\llap{-}300}\]
\[\small{\llap{-}400}\]
\[\small{\llap{-}500}\]
What mistakes did Benjamin 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
Answers
GPT-4o mini
To evaluate Benjamin's graph and identify potential mistakes, let's analyze the situation as described:
1. **X-axis Scaling**: The x-axis represents time in years, which is shown to scale from 0 to 8. The problem states that Benjamin wants to show the account's balance over the first 10 years, so a scaling from 0 to 8 is insufficient and does not cover the entire desired range. Therefore, the x-axis is indeed inappropriate.
2. **Y-axis Scaling**: The y-axis shows a scale from -600 to 600. However, since Benjamin starts with an investment of $400 and earns interest, the balance will never go negative, meaning the lower part of this scale is irrelevant. The range should start from 0 and extend to a value greater than the maximum balance after 10 years. Given the growth model, when \( T = 10 \):
\[
B = 400 \cdot 1.05^{10} \approx 400 \cdot 1.62889 \approx 651.56
\]
Thus, the y-axis should properly accommodate values from 0 up to at least that amount. This means the scaling from -600 to 600 is inappropriate.
3. **X-axis Label**: The label for the x-axis is stated as "time in years," which is correct, since it conveys the intended meaning of the data being represented.
4. **Y-axis Label**: The label for the y-axis is stated as "Balance in dollars," which is also correct and appropriately describes the data.
Based on the evaluation:
- The correct choices that indicate the mistakes Benjamin made when drawing the graph are **Choice A (Inappropriate scale for x-axis)** and **Choice B (Inappropriate scale for y-axis)**.
Thus, the answers are:
- **A**
- **B**
1. **X-axis Scaling**: The x-axis represents time in years, which is shown to scale from 0 to 8. The problem states that Benjamin wants to show the account's balance over the first 10 years, so a scaling from 0 to 8 is insufficient and does not cover the entire desired range. Therefore, the x-axis is indeed inappropriate.
2. **Y-axis Scaling**: The y-axis shows a scale from -600 to 600. However, since Benjamin starts with an investment of $400 and earns interest, the balance will never go negative, meaning the lower part of this scale is irrelevant. The range should start from 0 and extend to a value greater than the maximum balance after 10 years. Given the growth model, when \( T = 10 \):
\[
B = 400 \cdot 1.05^{10} \approx 400 \cdot 1.62889 \approx 651.56
\]
Thus, the y-axis should properly accommodate values from 0 up to at least that amount. This means the scaling from -600 to 600 is inappropriate.
3. **X-axis Label**: The label for the x-axis is stated as "time in years," which is correct, since it conveys the intended meaning of the data being represented.
4. **Y-axis Label**: The label for the y-axis is stated as "Balance in dollars," which is also correct and appropriately describes the data.
Based on the evaluation:
- The correct choices that indicate the mistakes Benjamin made when drawing the graph are **Choice A (Inappropriate scale for x-axis)** and **Choice B (Inappropriate scale for y-axis)**.
Thus, the answers are:
- **A**
- **B**