Time Model – T = aL + bG (AMC-inspired)

🧭 Problem (AMC credit)

Adapted from an AMC-style modelling problem.

A model to estimate the time \(T\) (in minutes) to hike to the top of a mountain on a trail is

\[ T = aL + bG \]

where:

\(L\) = length of the trail in miles
\(G\) = altitude gain in feet
\(a, b\) = constants (same for all trails)

The model says it will take 69 minutes to hike

a 1.5-mile trail with 800 ft gain, and
a 1.2-mile trail with 1100 ft gain.

Question: How many minutes does the model estimate for a trail that is 4.2 miles long and ascends 4000 ft?

Options: (A) 240 (B) 246 (C) 252 (D) 258 (E) 264

1️⃣ Form the equations

Substitute the two given situations into \(T = aL + bG\).

First trail

Time = 69, length = 1.5, gain = 800:

\[ 69 = 1.5a + 800b \quad \text{(Eq. 1)} \]

Second trail

Time = 69, length = 1.2, gain = 1100:

\[ 69 = 1.2a + 1100b \quad \text{(Eq. 2)} \]

So we have a simple system of two linear equations in two variables \((a,b)\).

2️⃣ Solve for \(a\) and \(b\)

Step A – subtract

Subtract (Eq. 2) from (Eq. 1):

\[ (1.5a - 1.2a) + (800b - 1100b) = 69 - 69 \] \[ 0.3a - 300b = 0 \] \[ a = 1000b \]

Step B – substitute

Put \(a = 1000b\) in Eq. 1:

\[ 69 = 1.5(1000b) + 800b = 1500b + 800b = 2300b \] \[ b = \frac{69}{2300} = 0.03 = \frac{3}{100} \] \[ a = 1000b = 1000 \times 0.03 = 30 \]

So the model becomes \(T = 30L + 0.03G\). In words: “30 minutes per mile + 0.03 minutes per foot climbed”.

3️⃣ Predict for the new trail

Now use \(L = 4.2\) and \(G = 4000\):

\[ T = 30(4.2) + 0.03(4000) \] \[ = 126 + 120 = 246 \]

So estimated time = 246 minutes.

Correct option: (B) 246

🧠 Intuition for laymen

The time depends on two things only: how far you walk (miles) and how high you climb (feet).
\(a=30\) means: even if the trail was flat, every mile would still take about 30 minutes.
\(b=0.03\) means: every extra foot of climbing adds about 0.03 minutes (≈ 1.8 seconds).
So a long + steep trail will obviously take more time — and our formula captures exactly that.

📝 Quick Quiz (based on this AMC idea)

Q1. Using the model you just derived, what is the time for a trail that is \(3.6\) miles long and climbs \(2000\) ft?

Use \(T = 30L + 0.03G\).

🧪 Extra Practice (Model Variation)

Q2. Suppose a different trail takes 100 minutes and is 2 miles long with 1000 ft gain. Using the same model \(T = 30L + 0.03G\), is the model overestimating or underestimating this trail?