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Precalculus 9-12 / Week 04 / Wednesday
3/6
Week 04 · Inverses Everywhere

Wednesday

Boathouse Lab: the tube winch
// The winch rule run backwards
⏱ about 20 min

Wednesday: Boathouse Lab, the Tube Winch

A grown-up stands by the real winch while the crew builds a small one on the dock bench.

Comet tapes one end of a string to a cardboard tube. "30 centimeters of string on the tube before we start. Now turn."

Wren turns once and measures the string pulled in. "40. Again. 50. Again. 60."

"Ten centimeters every turn," Comet says. "So string on the tube is 10 times turns, plus 30. What do you notice?"

"A straight rule," Wren says. "Which means it runs backwards cleanly."

Nova projects the table beside the tube. "Would you like a hint? Try the backwards question before you turn."

"70 centimeters," Comet calls. Wren writes the inverse, puts in 70, and says "4 turns." Four turns later, the tape agrees.

What you need

  • A cardboard tube, about 1 meter of string, tape, a tape measure or ruler, and your Boathouse Log.
  • A marker to mark one spot on the tube so you can count whole turns.
  • Scissors to trim the string. A flat table or bench.
Safety first
A grown-up is on the dock the whole time, and life vests are worn near the water.
The real winch is for grown-ups only. Your winch is a cardboard tube on a bench.
Cut string with scissors only, sitting down. Keep string away from necks and off the floor.
Nothing heavy is lifted alone. Ask for help moving the bench or the boat cradle.

Run the lab

  1. Tape the string to the tube and wind on 30 centimeters before you count. Mark the tube with a dot.
  2. Turn the tube one full turn, dot to dot. Measure the string now on the tube. Record it.
  3. Repeat for five turns. Write turns in one column and centimeters in the next.
  4. Find the change per turn. Write your rule: centimeters = (change per turn) × turns + start.
  5. Find the inverse by swapping and solving. Write it beside the rule.
  6. Pick a length. Use the inverse to predict the turns, then turn and measure to check.

The crew's readings

The crew's numbers are made up for the Boathouse. Yours depend on your tube, and that is the point of a lab.

Turns (x)String on the tube (cm)Backwards
030f⁻¹(30) = 0
140f⁻¹(40) = 1
250f⁻¹(50) = 2
360f⁻¹(60) = 3
470f⁻¹(70) = 4
580f⁻¹(80) = 5

The crew's rule: f(x) = 10x + 30. The inverse: f⁻¹(x) = 0.1x - 3.

Check both ways. f⁻¹(f(4)) = f⁻¹(70) = 4. And f(f⁻¹(70)) = f(4) = 70.

Both compositions give back the number you started with. That is the test for an inverse, and Thursday says why.

READ THE LAB LOG
  • Read the question.
  • Tap your answer.
The tube rule is f(x) = 10x + 30, string in centimeters after x turns. What is the inverse?
The lab table: f(0) = 30, f(1) = 40, f(2) = 50, f(3) = 60, f(4) = 70, f(5) = 80. What is f⁻¹(60)?
f(x) = 10x + 30 and f⁻¹(x) = 0.1x - 3. What is f(f⁻¹(70))?
f(x) = 10x + 30 and f⁻¹(x) = 0.1x - 3. What is f⁻¹(f(4))?
Comet calls out 70 centimeters. How many turns does the inverse rule predict? Type the number.
WHY THIS EXERCISEThe inverse answers the lab's backwards question before you turn the tube. The tape then checks it.
What the lab showsTrue or false?
The string on the tube grows by the same amount each turn.?
10 × 4 + 30 = 70?
To run the rule backwards, divide by 10 first and then subtract 30.?
f(f⁻¹(x)) gives back x for every length in the table.?
A thicker tube would change the start but not the change per turn.?
WHY THIS EXERCISEA lab turns a rule you were told into a rule you measured, then ran backwards yourself.
Draw your turns and centimeters as dots, turns across and centimeters up. Draw the line through them.

Careful lab work. Tomorrow the inverse gets its picture: a mirror across the line y = x.

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