Monday
Monday: Drops Against Time
The drip hose runs the length of the raised beds, one slow drop at a time into a measuring cup.
Comet holds the stopwatch. "Ten milliliters in the cup to start. Four more every second. What is our rule?"
"V equals 4t plus 10," Wren says. "Volume from time. What do you notice if I ask it backwards?"
"How long until the cup holds 50? Then 50 = 4t + 10, so t is 10 seconds," Comet says.
Nova projects a second rule beside the first. "Would you like a hint? Write the backwards rule once, for any volume."
"t equals V minus 10, all over 4," Comet says. "The same rule, run in reverse."
"An inverse," Wren says. "And look at the wet patch under the drip. Its radius is a square root of time."
"Then finding the time means undoing a square root," Comet says. "Squaring."
A rule and its reverse
The crew's cup rule is V(t) = 4t + 10, volume in milliliters after t seconds. It is the crew's own fit to their stopwatch readings.
Forwards: V(5) = 30. Backwards: when is V = 50? Solve 50 = 4t + 10, so t = 10.
Done once for any volume, the backwards rule is t = (V - 10)/4. In function form, V⁻¹(x) = 0.25x - 2.5.
A function and its inverse undo each other. Put 5 into V, get 30. Put 30 into V⁻¹, get 5 back.
The crew's drip table
Every row is the crew's own reading from Nova's log, made up for the Glass House. The inverse reads the table right to left.
| Seconds t | V(t) = 4t + 10 (mL) | Read backwards |
|---|---|---|
| 0 | 10 | V⁻¹(10) = 0 |
| 5 | 30 | V⁻¹(30) = 5 |
| 10 | 50 | V⁻¹(50) = 10 |
| 15 | 70 | V⁻¹(70) = 15 |
| 20 | 90 | V⁻¹(90) = 20 |
A solved problem to study: a square root in the rule
The wet patch under the drip grows slowly. The crew's model from their tape measure is r = √(4t + 1), radius in centimeters after t seconds.
When is the radius 5 cm? That asks for t under a square root: a radical equation.
- Start with √(4t + 1) = 5.
- Square both sides to undo the root: 4t + 1 = 25.
- Solve the linear equation: 4t = 24, so t = 6.
- Check in the original: √(4 × 6 + 1) = √25 = 5. It works.
Squaring undoes a square root, just as subtracting 10 and dividing by 4 undo the cup rule. The check at the end matters; tomorrow you see why.
- Read the question.
- Tap your answer.
- Read the question.
- Tap your answer.
| Statement | True or false? |
|---|---|
| V(t) = 4t + 10 gives the volume from the time. Its inverse gives the time from the volume. | ? |
| 4 × 10 + 10 = 50 | ? |
| (90 - 10) ÷ 4 = 20 | ? |
| Squaring both sides of √(4t + 1) = 5 gives 4t + 1 = 5. | ? |
| A function and its inverse undo each other. | ? |
Pick a number, run it through the front, then through the back. Did you get your number back?
Strong start. Tomorrow a radical equation gives an answer that fails the check, and you learn why.