When Logan left his house this morning, his cell phone was 24% charged and it then started to lose 3% charge for each hour thereafter. Write an equation for B,B, in terms of t,t, representing the charge remaining in Logan's battery, as a percentage, tt hours after Logan left his house.
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There are also ample opportunities to talk about the role of modeling here, touching on mathematical practice standard MP4. How reasonable is it that the output units are reported as percents? Does the model hold for all time? In particular, note that the model predicts that the percent charged grows linearly for all time, even beyond 100%!
If the task is done in small groups, different groups would likely use different representations in their solutions. Having groups present their answers could lead to a rich discussion on connecting different representations of functions.
Solutions
Solution: Finding and using equations
The battery charge of both devices can be modeled with linear functions. The wording describing the MP3 player suggests a linear function since it uses a constant rate of change. The table of values for the video game player shows a constant rate of change for the first 30 minutes. It is a reasonable assumption that the battery will continue to charge at the same rate. However, it is an assumption on our part. (Another possibility would be that as the battery charge approaches 100%, the rate of change decreases, but that would be much harder to model.)
The MP3 player charges at a rate of 12 percentage points every 15 minutes, which is equal to 0.8 percentage points per minute. If we let y be battery charge of the device (in percentage points) we have:
y=0.8t+40,
where t is measured in minutes.
We know that the video game player is initially 20% charged and from the table we see that the charge increases by an additional 12 percentage points every 10 minutes, or 1.2 percentage points per minute. So for this function we get:
y=1.2t+20.
Sam’s family is planning to leave the house 60 minutes after Sam started charging his devices. We are looking for the charge when t=60:
MP3 player: y=0.8⋅60+40=88 % charged
video game player: y=1.2⋅60+20=92 % charged
To answer this question, we need to find the values of t for which each function has output value 100.
MP3 player: Solving 100=0.8t+40 for t we have, t=75 minutes.
video game player: Solving 100=1.2t+20 for t we have t=67 minutes.
So if Sam’s family could wait just 15 more minutes, Sam could have both devices fully charged for the car trip.
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