Ask a New Question
Menu

Clemence studies a sample of a chemical element called Einsteinium-253, which naturally loses its mass over time. She modeled the relationship between the time

\[T\] since she started studying the sample (in days) and the sample's mass
\[M\] (in grams) as

\[M=600\cdot0.5^{^{\scriptsize\dfrac{T}{20.5}}}\].
She wanted to create a graph to show the sample's decay in the first month. Here is her work:
An exponential function decreases as x increases on an x y coordinate plane. The x axis scales by three from zero to thirty. The y axis scales by two hundred from zero to two thousand. The x axis is labeled Mass in grams. The y axis is labeled Time in days. The graph starts at the y axis at (zero, six hundred) passing through the points around (twelve, four hundred) and (thirty, two hundred).
\[\small{3}\]
\[\small{6}\]
\[\small{9}\]
\[\small{12}\]
\[\small{15}\]
\[\small{18}\]
\[\small{21}\]
\[\small{24}\]
\[\small{27}\]
\[\small{200}\]
\[\small{400}\]
\[\small{600}\]
\[\small{800}\]
\[\small{1000}\]
\[\small{1200}\]
\[\small{1400}\]
\[\small{1600}\]
\[\small{1800}\]
What mistakes did Clemence 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) Clemence didn't make any mistake
E
Clemence didn't make any mistake

1 answer

To evaluate Clemence's work, let's analyze the details provided regarding the graph of the equation \( M = 600 \cdot 0.5^{\frac{T}{20.5}} \).

  1. Understanding the Function: Clemence's equation represents the mass \( M \) of Einsteinium-253 over time \( T \). The initial mass is 600 grams, and it halves every 20.5 days.

  2. Domain and Range:

    • The domain (time \( T \)) in this situation would be from 0 to 30 days (one month).
    • The mass \( M \) starts at 600 grams and decreases. After 30 days, it can be calculated using the formula: \[ M(30) = 600 \cdot 0.5^{\frac{30}{20.5}} \approx 600 \cdot 0.5^{1.4634} \approx 600 \cdot 0.348 = 208.8 \text{ grams} \]
    • This indicates that at \( T = 30 \) days, the mass will be around 208.8 grams, not the stated 200 grams for \( (30, 200) \).

  3. Axes Labelling:

    • The x-axis in the equation should be labeled as "Time in days" rather than "Mass in grams."
    • The y-axis should be labeled as "Mass in grams" instead of "Time in days."

  4. Scale Issues:

    • The x-axis scales from 0 to 30 days; this seems appropriate.
    • The y-axis scales from 0 to 2000 grams. Given that the function goes from 600 grams down to approximately 200.8 grams over 30 days, using a scale that goes to 2000 grams seems excessive and non-representative for the actual values.

Based on this analysis, Clemence made the following mistakes:

  • (Choice A): Inappropriate scale for the \( x \)-axis - Incorrect (The scale seems appropriate)
  • (Choice B): Inappropriate scale for the \( y \)-axis - Correct (The scale goes to 2000 grams, which is excessive)
  • (Choice C): Wrong/missing label for the \( x \)-axis - Correct (Should be "Time in days")
  • (Choice D): Wrong/missing label for the \( y \)-axis - Correct (Should be "Mass in grams")
  • (Choice E): Clemence didn't make any mistake - Incorrect

Therefore, the correct choices are B, C, and D.