Question

Ashlynn models the relationship between the temperature
\[C\] of a hot cup of tea (in degrees Celsius) and time
\[T\] since the tea was poured into the cup (in minutes) as
\[C=20+20\cdot 0.88^T\].
She wanted to graph the relationship over the first half hour. Here is her work:
An exponential function decreases as x increases on an x y coordinate plane. The x axis scales by five from zero to thirty. The y axis scales by five from zero to forty five. The x axis is labeled Time in minutes. The y axis is labeled Temperature in degrees Celsius. The graph starts at the y axis at (zero, forty) passing through the points around (five, four thirty) and (ten, twenty five). The asymptote is at y equals twenty.
\[\small{5}\]
\[\small{10}\]
\[\small{15}\]
\[\small{20}\]
\[\small{25}\]
\[\small{5}\]
\[\small{10}\]
\[\small{15}\]
\[\small{20}\]
\[\small{25}\]
\[\small{30}\]
\[\small{35}\]
\[\small{40}\]
What mistakes did Ashlynn 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) Ashlynn didn't make any mistake
E
Ashlynn didn't make any mistake

Answer 1

To evaluate Ashlynn's work and identify the mistakes, let’s analyze the description of the graph in relation to the given function \(C = 20 + 20 \cdot 0.88^T\) and the specific points mentioned:

1. **Function Behavior**: The function is exponential decay, starting from \(C = 40\) when \(T = 0\) and approaching \(C = 20\) as \(T\) increases. The graph should start at (0, 40) and decrease towards the horizontal asymptote at \(C = 20\).

2. **Points Mentioned**:
- At \(T = 0\), \(C = 40\) (correct).
- At \(T = 5\): \(C = 20 + 20 \cdot 0.88^5\) which approximates to about \(C \approx 30.57\) (not 430).
- At \(T = 10\): \(C = 20 + 20 \cdot 0.88^{10}\) approximates to about \(C \approx 25.18\) (also inconsistent with 25 mentioned).

Given these observations, let’s analyze the choices:

- **Choice A**: Inappropriate scale for \(x\)-axis - **Not applicable**. The scale of 0 to 30 (in increments of 5) is appropriate for the time range of interest (first half hour).

- **Choice B**: Inappropriate scale for \(y\)-axis - **Applicable**. The scale from 0 to 45 is fine, but the actual values of the function are much lower than stated for particular points. The critical points should not go up to 45.

- **Choice C**: Wrong/missing label for \(x\)-axis - **Not applicable**. The x-axis is labeled correctly as "Time in minutes".

- **Choice D**: Wrong/missing label for \(y\)-axis - **Not applicable**. The y-axis is labeled correctly as "Temperature in degrees Celsius".

- **Choice E**: Ashlynn didn't make any mistake - **Not applicable**. Ashlynn made mistakes in reporting specific point temperatures.

In conclusion, the relevant mistake made by Ashlynn is:

**(Choice B) Inappropriate scale for y-axis.**