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Algebra 2 9-12 / Week 07 / Wednesday
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Week 07 · Logarithms

Wednesday

Glass House Lab: the cooling jug
// The exponent, found backwards
⏱ about 20 min

Wednesday: Glass House Lab, the Cooling Jug

A grown-up pours warm water into the clear jug and sets it on the potting bench. Wren slides the thermometer in.

"Room is 18 degrees," he says. "Water reads 42. Start the stopwatch."

Every minute Comet calls the time and Wren reads the number. 37.2. 33.4. 30.3.

"What do you notice?" Wren asks. "The drops get smaller each minute."

Nova projects the readings beside the room line. "Would you like a hint? Look at the gap above 18, not the reading."

"The gap goes 24, then 19.2, then about 15.4," Comet says. "Each one is 0.8 of the last."

"So the gap is exponential," Wren says. "Which means a logarithm tells us when it halves."

What you need

  • A clear jug, a kitchen thermometer, a stopwatch or kitchen timer and your Glass House Log.
  • Warm water from the tap, poured by a grown-up. Not boiling, just warm to the touch of the jug.
  • A steady bench or table in a room whose temperature you measure first.
Safety first
A grown-up handles the warm water and pours it. You never carry or pour it.
Keep the jug on a flat surface away from the edge. Wipe up any spill at once so no one slips.
Wait for the thermometer reading to settle before writing it down. Never put the thermometer in your mouth.

Run the lab

  1. Measure the room temperature and write it at the top of your table.
  2. The grown-up pours the warm water. Put the thermometer in and start the stopwatch at the first reading.
  3. Read the thermometer every minute for six minutes. Record each reading.
  4. For each row, subtract the room temperature to find the gap.
  5. Divide each gap by the gap before it. Write the factor in the last column.
  6. Find your middle factor. That is your crew's cooling model for this jug.

The crew's readings

The crew's numbers are made up for the Glass House. Yours will differ, and that is the point of a lab.

MinuteReadingGap above roomGap ÷ last gap
04224(start)
137.219.20.8
233.415.40.8
330.312.30.8
427.89.80.8
525.97.90.8

The crew's model: gap = 24 × 0.8ᵗ, so the reading is 18 + 24 × 0.8ᵗ. The 0.8 is their own fit, not a law.

When is the gap half of 24? Solve 24 × 0.8ᵗ = 12. Divide: 0.8ᵗ = 0.5. So t = log 0.5 ÷ log 0.8 ≈ 3.1 minutes.

A quarter of the gap takes about 6.2 minutes, twice as long. Each halving takes the same time in this model.

READ THE COOLING MODEL
  • Read the question.
  • Tap your answer.
The room is 18 degrees and the soil starts at 42 degrees. The crew's model shrinks the gap by a factor of 0.8 each minute. What is the reading after 3 minutes? (Round to 1 place.)
The room is 18 degrees and the soil starts at 42 degrees. The crew's model shrinks the gap by a factor of 0.8 each minute. What is the reading after 5 minutes? (Round to 1 place.)
The gap starts at 24 degrees and shrinks by a factor of 0.8 each minute. After how many minutes is the gap 12? (Round to 1 place.)
The gap starts at 24 degrees and shrinks by a factor of 0.8 each minute. After how many minutes is the gap 6? (Round to 1 place.)
In the crew's model the reading is 18 + 24 × 0.8ᵗ. What is the reading after 1 minute? Type the number.
WHY THIS EXERCISEThe exponential part is the gap, not the reading. Adding the room temperature back is the last step every time.
StatementTrue or false?
The gap above room temperature shrinks by the same factor each minute in the crew's model.?
42 - 18 = 24?
The reading itself, not the gap, is what gets multiplied by 0.8.?
Halving the gap twice takes about twice as long as halving it once.?
The 0.8 is a law that every jug in every room must follow.?
WHY THIS EXERCISELogarithms answer "how long" for anything that shrinks or grows by a steady factor.
Draw your six readings as dots, minute across and degrees up. Draw a flat line at room temperature underneath.

Careful lab work. Tomorrow the logarithm gets its own graph, a mirror of the exponential curve.

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