"Open House planters," Comet says, unrolling a sheet of cardboard. "Square ones. This one needs 36 square decimeters of base."
"Area from side is s squared," Wren says. "You want side from area. What do you notice?"
"The inverse. Side is the square root of area. Root of 36 is 6, so 6 by 6."
Nova hovers over the sheet. "Would you like a hint? A side is never negative, so only one root counts."
"Same as the drip," Comet says. "Volume from time, time from volume. Area from side, side from area."
"Every rule you use forwards, someone needs backwards," Wren says. "That is why inverses matter."
"And every time there is a square, there is a root," Comet says, and picks up the scissors.
| Forwards rule | Inverse | A Glass House use |
|---|---|---|
| V(t) = 4t + 10 | V⁻¹(x) = 0.25x - 2.5 | seconds to wait for a volume |
| A = s² (side 0 or more) | s = √A | the side of a square planter from its base area |
| V = s³ | s = ∛V | the edge of a cube box from its volume |
| r = √(4t + 1) | t = (r² - 1)/4 | seconds until the wet patch has a radius |
A square planter base of 36 square decimeters has side √36 = 6 dm. One of 81 has side 9 dm. The negative root is not a side.
A cube box of 27 cubic decimeters has edge ∛27 = 3 dm. Written with a rational exponent, that is 27^(1/3).
The wet patch rule run backwards: r = √(4t + 1) becomes t = (r² - 1)/4. For r = 9, t = 20 seconds.
| Statement | True or false? |
|---|---|
| To solve a radical equation, square both sides, solve, then check in the original. | ? |
| Squaring both sides can create an extraneous solution. | ? |
| 8^(2/3) = 4 | ? |
| A function has an inverse function only when it is one-to-one. | ? |
| The graph of y = ∛x stops at x = 0. | ? |
Terrific week. You can undo a root, build an inverse and tell when one exists. Tomorrow is Garden Day.