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Day 7 — What you're building today

You'll write your bot's decay equation in slope-intercept form and calculate two specific times: where the math line flattens at 20% (the power-budget floor) and where the mathematical line crosses P = 0 (the true x-intercept). Both become marked points on your graph in Section 3.

Section 1 of 3

From Formula to Equation

Your P(t) formula is already linear — let's write it in slope-intercept form.

The Endurance Decay Formula

P(t) = 100 − (20 − E) × 1.5 × t

P(t) = motor power % · = time in minutes · E = your Endurance stat

🔍 What does each part mean?

P(t) = 100 − (20 − E) × 1.5 × t

P(t)

Power right now

How much power your bot has at any moment in the match. Starts high, goes down.

100

Starting power

Every bot starts at 100% power — no matter what your E value is.

(20 − E) × 1.5

Power lost per minute

Higher E → smaller (20−E) → smaller decay rate. A slope of −27 means your bot loses 27% power every minute. A slope of −6 means it loses only 6% per minute. This is your slope.

Time (minutes)

How long the match has been running. Multiply time × decay rate to get total power lost.

🚫 Stop — most students get this wrong

Higher Endurance does NOT mean more decay. The subtraction (20 − E) flips it: a bigger E makes the decay rate smaller.

❌ Wrong: r = E × 1.5

E = 18 → r = 27

High endurance = fastest death?
That's backwards.

✓ Correct: r = (20 − E) × 1.5

E = 18 → r = (20−18)×1.5 = 3

High endurance = slow drain.
Makes sense.

Rule: Higher E → smaller (20−E) → smaller decay rate → longer survival.

⚡ P(t) is a power bar — watch it drain over the match

POWER

t = 0 min (100%) t = 2 min t = 3 min ← match ends

P(t) = the height of the bar at any moment in time. In Velocity Arena, this power % stops draining at 20% — the game engine holds the math line there. A higher E stat slows the drain.

⚠️

Two different floors — don't mix them up

P(t) is your power budget (a %), and the math line holds at 20%. But your bot's wheels also have a hard minimum speed, set by your SPEED stat. Because that minimum is higher than 20% of your motor power, your wheels bottom out — and the lights turn red — earlier than the 20% mark. So the 20% point is where the math line flattens, not the moment your bot first feels slow.

This is already a linear equationLinear equationAn equation that makes a straight line when graphed. Power goes down at a steady rate — no curves.. You've seen them in math class written as y = mx + b. Let's rewrite P(t) in that exact form — same equation, different look.

Rewrite in slope-intercept form — click to reveal each step

1

P(t) = 100 − (20 − E) × 1.5 × t

Start with the original formula. Notice: 100 is at the front, and the variable term (with t) is at the end.
2

P(t) = −[(20 − E) × 1.5] × t + 100

Group the coefficientCoefficientThe number multiplied by a variable. In 3x, the coefficient is 3. Here, everything multiplied by t is the coefficient.: the term being multiplied by t is negative. We flip the subtraction into a negative slopeSlope (m)How steep the line is. Negative slope = line goes down. Here it means power decreases over time., and move the constant +100 to the end.
3

y = m · x + b

Match to y = mx + b: m = −(20 − E) × 1.5 (the slope), b = 100 (the y-intercept).
y = P(t) and x = t.

🗺 How P(t) maps to y = mx + b

y = m · x + b ← standard form

↕ ↕ ↕ ↕

P(t) = −(20−E)×1.5 · t + 100 ← your decay equation

m (slope) = how fast power drops per minute — "a slope of −18 means your bot loses 18% every minute"

b (y-intercept) = starting power (always 100)

x = time t in minutes

y = power P(t) as a %

Check for Understanding

In your equation P(t) = mx + 100, the y-intercept b = 100. What does 100 represent in the context of the game?

Section 2 of 3

Two Times on Your Graph

Your equation has two important moments: when the game locks power at 20%, and where the line mathematically hits zero. You'll calculate both — and they are not the same number.

Look at this graph before reading anything.

One point happens in the game. One point only exists mathematically. Which comes first?

20% Floor Time: Solve P(t) = 20

Set P(t) = 20 and solve for t — when does the game engine lock power?

20 = m · t + 100

−80 = m · t ← subtract 100 from both sides to get the t-term alone

t = −80 / m ← divide both sides by m to isolate t

m is a negative number — dividing −80 by a negative gives a positive result, so t > 0

t = 80 / ((20 − E) × 1.5) ← 20% floor time

Your calculation

E = Enter your E value to see the steps

⏱ Visual: Does your bot survive the 3-minute match?

Match

3 min match

3 min

Your bot

← enter E above

0 min

1 min

2 min

3 min

4 min

5 min

Your bot's 20% floor time is t = ? minutes. The match lasts 3 minutes. Does your bot hit the 20% floor before time runs out?

💬 Word bank — click to add to your answer

my bot survives dies before the match ends hits floor after 3 minutes hits floor before 3 minutes I should increase E no problem

0 / 10 minimum

You've found the 20% floor time. Now look at your graph — the solid line stops there. The game engine holds it.

But this is a straight line with a slope. Mathematically, it doesn't stop at 20%. If the game rule didn't exist, where would the line end?

At the x-interceptX-interceptWhere a line crosses the horizontal axis. Set y = 0 to find it., the graph touches the horizontal axis. On this graph, that axis is P = 0 — zero power. Setting P(t) = 0 finds that exact point.

True X-Intercept: Solve P(t) = 0

Same process — different target. Set P(t) = 0 instead of 20.

0 = m · t + 100

−100 = m · t ← subtract 100 to get the t-term alone (same move as before)

t = −100 / m ← divide by m to isolate t (negative ÷ negative = positive)

t = 100 / ((20 − E) × 1.5) ← true x-intercept (P = 0)

This number is always larger than your floor time — the game stops the line at 20% before it ever reaches 0.

E=2 vs E=16 — Dramatic Contrast

E Value	Slope (m) = power loss/min	20% Floor Time	True X-Intercept (P=0)	Match result
E = 2	−27 loses 27%/min	≈ 2.96 min ⚠	≈ 3.70 min	Hits floor before match ends
E = 8	−18 loses 18%/min	≈ 4.44 min	≈ 5.56 min	Survives full match
E = 16	−6 loses 6%/min	≈ 13.3 min	≈ 16.7 min	4× the match length ✓✓
Your E = ?	—	—	—	Enter your E above

Proportional Check

You just calculated two times for your bot:

t_floor — when power hits the 20% floor (game stops draining here)

t_xi — where the line mathematically reaches P = 0 (the true x-intercept)

If these two times are always connected by a fixed ratio, you only ever need to calculate one of them — you can get the other instantly by multiplying.
Divide t_xi by t_floor using your numbers above and see if that's true.

        txi ÷ tfloor
        =
        __
        ÷
        __
        =
        
      

Section 3 of 3

Your Personalized Equation

Build your decay equation from your stat card and reflect on what it means.

Enter your actual E value from your stat card

E = From your stat card

My Decay Equation

y = ___x + 100

Power Decay Graph

Actual gameplay

Mathematical line (no game rule)

Reflection 1

What does your slope tell you about how quickly your bot loses power?

My slope is ___, which means my bot loses ___% of power every minute.

The values above are auto-filled from your E. Confirm by typing a brief explanation:

💬 Word bank — click to add to your answer

loses power quickly loses power slowly steep slope gentle slope drops % per minute higher E = less loss negative slope

0 characters

Reflection 2 · Predict & Compare

Your bot's 20% floor time is ___ min. Which improves survival more — raising E from 2 to 4, or from 16 to 18? Calculate both 20% floor times and explain why one change matters more.

E : 2 → 4

t = 80 ÷ ((20 − E) × 1.5)

E = 2 → min

E = 4 → min

Gain: —

E : 16 → 18

t = 80 ÷ ((20 − E) × 1.5)

E = 16 → min

E = 18 → min

Gain: —

💬 Word bank — click to add to your answer

larger change in survival time smaller denominator decay rate drops by half from 27 to 24 from 6 to 3 dividing by a much smaller number

0 / 15 minimum

Reflection 3 · Apply Both Calculations

A teammate has E = 2. Write their decay equation, then find both their 20% floor time AND their true x-intercept (P = 0).

Hint: m = −(20 − 2) × 1.5 = −27. Floor time: solve P = 20. X-intercept: solve P = 0. They are different numbers.

Their equation

20% Floor Time (min)

True X-Intercept, P=0 (min)