Wednesday
Wednesday: Boathouse Lab, Measuring the Oars
A grown-up watches from the dock while the crew spreads graph paper on the workbench.
"Set the oars at 75°," Comet says, reading the protractor. "Now a 10 centimeter line from the hinge along the second oar."
Wren drops a line straight down from its tip to the bench edge and measures. "9.7 centimeters high."
"Divided by 10 is 0.97," Comet says. "Sin 75° from the formula was about 0.97. What do you notice?"
"The ruler agrees with the algebra," Wren says. "Now 15°. The height should be tiny."
Nova projects the three triangles in a row. "Would you like a hint? The 10 centimeter line is the hypotenuse every time."
"2.6 centimeters," Comet reads. "About 0.26. The formula said 0.26. Close enough for cardboard."
What you need
- Two strips of cardboard about 20 centimeters long, a paper fastener or a pin for the hinge, and tape.
- A protractor, a ruler, a sheet of graph paper, a pencil and your Boathouse Log.
- Scissors for the cardboard. A flat table or bench.
Cut cardboard with scissors only, sitting down. A grown-up pushes in the paper fastener or pin.
Keep the pin's point covered with tape once the hinge is made. Pins stay off the floor.
Nothing heavy is lifted alone. Real oars stay on their rack.
Run the lab
- Cut two cardboard strips with scissors. A grown-up joins one end of each with the fastener so they swing.
- Tape the first oar along the bottom line of the graph paper. Set the second oar 75° above it with the protractor.
- Mark a point 10 centimeters from the hinge along the second oar. Drop a line straight down to the bottom line.
- Measure the height of that line. Divide by 10. That is your measured sin 75°.
- Repeat at 15° and at 105°. For 105° the point lands past the hinge, but the height still counts.
- Compare each measured value with the exact value from the formula and with its decimal.
The crew's readings
The crew's numbers are made up for the Boathouse. Yours depend on your cardboard and your ruler, and that is the point of a lab.
| Angle | Line from hinge (cm) | Height (cm) | Height ÷ line | Exact sine |
|---|---|---|---|---|
| 15° | 10 | 2.6 | 0.26 | (-√2 + √6)/4 |
| 75° | 10 | 9.7 | 0.97 | (√2 + √6)/4 |
| 105° | 10 | 9.7 | 0.97 | (√2 + √6)/4 |
Each measured ratio sits within a few hundredths of the exact value. A ruler on cardboard is not perfect, but the formula holds.
For 105°, the height matched 75° exactly, because sin 105° = sin(180° - 75°) = sin 75°. Week 5's reflection rule, seen on the bench.
- Read the question.
- Tap your answer.
The 10 cm line from the hinge sits at 75°. How high is its tip above the bench edge? Round to 1 decimal place.
The 10 cm line sits at 15°. How high is its tip? Round to 1 decimal place.| What the lab shows | True or false? |
|---|---|
| The 10 cm line from the hinge is the hypotenuse of each triangle. | ? |
| Height ÷ line gives the cosine of the angle. | ? |
| The measured ratio at 75° was close to 0.97. | ? |
| sin 105° equals sin 75° because 105° = 180° - 75°. | ? |
| A ruler on cardboard gives the exact surd value. | ? |
Careful lab work. Tomorrow you prove the addition formula from the picture Nova projected.