Week 06 · Systems of Equations and Inequalities

Wednesday

Loft Lab: the catch-up race

// Two carts meet, and a region of good builds

⏱ about 20 min

Wednesday: Loft Lab, the Catch-Up Race

Comet sets the two planks side by side, both gentle, one a little steeper. "Lab day. Catch-up race."

"Cart A starts at the 40 mark on the gentle plank," Wren says. "Cart B starts at zero on the steeper one."

Comet rushes to release both. "Wait. Same moment, or the timing is useless." She counts down. "Three, two, one."

"What do you notice?" Wren asks, stopwatch in hand.

"B catches A right around 4 seconds, near the 120 mark. Just like the lines said."

Nova projects the two lines and a blinking dot at the crossing. "Would you like a hint? Check the readings each second."

"A: 60, 80, 100, 120. B: 30, 60, 90, 120," Wren reads. "Equal at 4. Our engineer, run your own race."

What you need

Two long boards side by side on low supports, one tilted slightly more than the other, towels at the ends.
Two small toy carts or wheeled toys, and masking tape marks every 10 centimeters on each board.
A stopwatch, two helpers to call the marks, and your Loft Log with graph paper.

Safety first

A grown-up sets up both boards and checks the supports cannot tip. Keep feet and fingers clear of rolling carts.
Keep both tilts gentle so the carts roll, never fly. Towels at the ends stop them.
Release the carts by letting go, never by pushing hard or throwing.

Run the lab

Place cart A at the 40 mark on the gentler board and cart B at the 0 mark on the steeper one.
Count down and release both carts at the same moment. Start the stopwatch as you let go.
Have each helper call the mark their cart passes every second. Write both lists side by side.
Plot both carts on one grid, seconds across and centimeters up. Draw a line through each set of dots.
Write a rule for each cart. Solve the system by substitution to predict the meeting time.
Compare the predicted meeting point with the second when the readings were equal.

The crew's lab log

The crew's own readings, in centimeters past the start line. Every number is their made-up Loft data.

Seconds	Cart A (y = 20x + 40)	Cart B (y = 30x)	Gap
0	40	0	40
1	60	30	30
2	80	60	20
3	100	90	10
4	120	120	0
5	140	150	-10

$A grid from 0 to 20 with the two cart lines and their readings as dots, drawn in tens of centimeters.$

READ THE CREW'S RACE

Read the question.
Tap your answer.

The crew's carts followed y = 20x + 40 and y = 30x. Which point is the meeting point?

Cart A read 60 at 1 second and 100 at 3 seconds. Which rule fits cart A?

In a second race A started at the 60 mark: y = 20x + 60 and y = 30x. Where do they meet?

In the crew's race, how many centimeters did the gap close each second? Type the number.

WHY THIS EXERCISEThe gap closes at the difference of the two rates, 30 - 20. That is why the meeting time is 40 ÷ 10.

Statement	True or false?
Releasing both carts at the same moment keeps the readings fair.	?
(4, 120) is a solution of y = 20x + 40 and y = 30x.	?
(5, 140) is a solution of y = 20x + 40 and y = 30x.	?
After the meeting point, cart B is ahead.	?
30 × 5 - (20 × 5 + 40) = 10	?

WHY THIS EXERCISEThe lab shows the crossing as a moment in time: before it A leads, after it B leads.

Try it

Give cart A a head start of 60 instead of 40 and race again. Predict the new meeting time first.
Keep both boards gentle and the towels in place.

Draw your race graph with both lines, labeled axes, a scale and the meeting point circled.

Loft Log: record both carts' readings, both rules and your predicted and observed meeting times.Did your observed meeting time match the prediction? Write one reason it might differ.Write what would happen to the meeting point if cart B rolled only 20 a second too.

Fine lab work, engineer. Tomorrow the crew turns build limits into inequalities and shades the region of good builds.

Grown-up checked my answers

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