The three finalists in a contest are brought to the centre of a large, flat field. Each is given a meter stick, a compass, a calculator, a shovel, and (in a different order for each) the following three displacements: 72.4m, 32? east of north, 57.3m, 36? south of west, 17.8m straight south. The three displacements lead to the point where the keys to a new Porsche are buried. Two contestants start measuring immediately, but the winner first calculates where to go. What does he calculate?

Answer

How the Contest Winner Found the Porsche Keys Using Vector Addition

How the Contest Winner Found the Porsche Keys

Using vector addition and basic trigonometry, the winner calculates the final displacement resulting from three separate movements to find the precise location of the buried keys.

Given Displacements:

Vector A: 72.4 m, at 32° east of north
Vector B: 57.3 m, at 36° south of west
Vector C: 17.8 m straight south

Step 1: Resolve Each Vector into Components

We break each vector into north-south (y) and east-west (x) components.

Vector A (32° east of north):

A_x = 72.4 × sin(32°) ≈ 38.4 m (east)
A_y = 72.4 × cos(32°) ≈ 61.3 m (north)

Vector B (36° south of west):

B_x = −57.3 × cos(36°) ≈ −46.3 m (west)
B_y = −57.3 × sin(36°) ≈ −33.7 m (south)

Vector C (Straight south):

C_x = 0 m
C_y = −17.8 m

Step 2: Add All Components

Sum all x and y components to find total displacement:

Horizontal (East-West) Total:

x_total = 38.4 + (−46.3) + 0 = −7.9 m (west)

Vertical (North-South) Total:

y_total = 61.3 + (−33.7) + (−17.8) = 9.8 m (north)

Step 3: Calculate Resultant Displacement

Use the Pythagorean Theorem to get the magnitude:

R = √(x² + y²) = √[(−7.9)² + (9.8)²] ≈ √(62.4 + 96.0) ≈ √158.4 ≈ 12.6 m

Direction (θ west of north):

θ = tan⁻¹(|x| / y) = tan⁻¹(7.9 / 9.8) ≈ 38.4°

✅ Final Displacement: 12.6 meters at 38.4° west of north

This is the calculated direction and distance the winner walks directly — while the others take the long route measuring. Smart thinking wins!

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