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Precalculus 9-12 / Week 09 / Dock Day
6/6
Week 09 · Ellipses and Hyperbolas

Dock Day

Dock Day: ovals of every shape
// The chalk oval on the lawn
⏱ about 20 min

Dock Day: Ovals of Every Shape

Saturday the pins, the string and a stack of cardboard sheets come home with Comet.

"One string, many ovals," she says. "Keep the string the same and move the pins. Watch the shape change."

"Then measure," Wren says. "Width, height and the pin gap. Check b² = a² - c² every time."

Nova dims her light. "Would you like a hint? Try the pins touching, then as far apart as the string allows."

"Touching pins make a circle," someone says. "Pins at the ends make a flat line. Everything else is in between."

"Record them all in the log," Comet says. "That table is the whole week in one page."

  1. An ellipse: the sum of distances to two foci is constant, the string length 2a. The foci are inside, at (±c, 0).
  2. x²/a² + y²/b² = 1 with b² = a² - c², derived by squaring the distance equation twice.
  3. A hyperbola: the difference of distances is constant, 2a. Its equation is x²/a² - y²/b² = 1 with c² = a² + b². The asymptotes are y = ±(b/a)x.
  4. A circle is an ellipse with its foci together. Complete the square to read its center and radius.
  5. A parabola has one focus and a directrix. Only one variable is squared.
◇ FAMILY OVALS
Tie a 10 centimeter string between two pins on cardboard. Start with the pins 8 centimeters apart and draw the oval.
Move the pins to 6 centimeters apart, then 2, then touching. Draw an oval each time with the same string.
Measure each oval's width and height. Halve them for a and b, halve the pin gap for c, and check b² = a² - c².
Trade roles so everyone sets pins, everyone draws and everyone measures. Keep one table for all the ovals.
For a grown-up
The grown-up sets and removes the pins. Pins stay in a lid between ovals and never on the floor.
A measured b² that misses a² - c² by a little is a good moment. Measure again and ask where the pencil wandered.
The ovals and their sizes are the family's own measurements, not facts about any real object.
DOCK DAY CHECK
  • Read the question.
  • Tap your answer.
A family oval is 12 centimeters wide, so a = 6. How long is the string tied between the pins?
An ellipse centered at the origin with half-axes 5 across and 4 up, with its two foci marked on the longer axisA family oval has the equation x²/25 + y²/16 = 1. Where are its two pins, the foci?
With a 10 centimeter string, the pins are moved to 6 centimeters apart, so c = 3. What is b? Type the number.
WHY THIS EXERCISESame string, closer pins, taller oval. The formula predicts the height before you draw it.
Draw your family's ovals one on top of another, all with the same width. Label each pin gap.
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