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Zorluk: OrtaDirection and Distance Test

A wildlife biologist tracking a radio-collared leopard starts at a research outpost and drives 10 km10\text{ km} straight East. She then turns 45∘45^{\circ} to her left and drives 102 km10\sqrt{2}\text{ km} along a dirt trail. Next, she turns 90∘90^{\circ} to her right and travels 42 km4\sqrt{2}\text{ km} through the brush. Finally, she turns 135∘135^{\circ} to her left and drives 1 km1\text{ km} straight North to locate the leopard. What is the shortest straight-line distance, in kilometers, between the research outpost and the leopard's final location?

Cevap: 25 km

Cevap

25
The correct answer is found by tracking the Cartesian coordinates after each vector movement. By breaking diagonal distances into their horizontal and vertical components, the final position is located exactly at (24,7)(24, 7). Using the distance formula 242+72\sqrt{24^2 + 7^2}, the shortest direct distance from the origin is 25 km25\text{ km}.

Adım Adım Çözüm

1
Establish a coordinate system with the starting point at (0,0)(0,0) and translate the first movement of 10 km10\text{ km} East.
Current position is (10,0)(10, 0).
East corresponds to the positive x-axis direction.
2
Determine the new heading after turning 45∘45^{\circ} left from East, and calculate the displacement for 102 km10\sqrt{2}\text{ km}.
New heading is North-East. Displacement is (10,10)(10, 10), making the current position (20,10)(20, 10).
A 45∘45^{\circ} left turn from East (0∘0^{\circ}) results in 45∘45^{\circ} (NE). The vector components are d×cos⁡(45∘)d \times \cos(45^{\circ}) and d×sin⁡(45∘)d \times \sin(45^{\circ}).
3
Determine the new heading after turning 90∘90^{\circ} right from North-East, and calculate the displacement for 42 km4\sqrt{2}\text{ km}.
New heading is South-East. Displacement is (4,−4)(4, -4), making the current position (24,6)(24, 6).
A 90∘90^{\circ} right turn from NE (45∘45^{\circ}) results in SE (−45∘-45^{\circ}). The vector components are d×cos⁡(−45∘)d \times \cos(-45^{\circ}) and d×sin⁡(−45∘)d \times \sin(-45^{\circ}).
4
Determine the new heading after a 135∘135^{\circ} left turn from South-East, and calculate the displacement for 1 km1\text{ km}.
New heading is North. Displacement is (0,1)(0, 1), making the final position (24,7)(24, 7).
A 135∘135^{\circ} left turn (positive rotation) from SE (−45∘-45^{\circ}) yields −45∘+135∘=90∘-45^{\circ} + 135^{\circ} = 90^{\circ} (North).
5
Apply the Pythagorean theorem to find the straight-line distance from the origin (0,0)(0,0) to the final coordinates (24,7)(24, 7).
The shortest distance is 25 km25\text{ km}.
The straight-line distance between two points on a Cartesian plane is given by (x2−x1)2+(y2−y1)2\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.

Anahtar Kavram

Vector displacement, angular turns mapping to cardinal/ordinal directions, and the Pythagorean theorem.
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Konular

Direction and Distance Test

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