A botanist leaves her base camp to collect soil samples in a dense forest. She first walks… — State PSC Exam

A botanist leaves her base camp to collect soil samples in a dense forest. She first walks $35\text{ m}$ straight East. She then turns $135^{\circ}$ to her left and walks $20\sqrt{2}\text{ m}$ . Finally, she turns $135^{\circ}$ to her left again and walks $12\text{ m}$ . What is the shortest straight-line distance, in meters, between her current position and the base camp?

Cevap: 17 m

Cevap

The shortest straight-line distance between the botanist's current position and the base camp is 17 meters.

By breaking each movement into its horizontal and vertical components on a Cartesian plane, the final position is found to be 15 meters East and 8 meters North of the origin. The shortest direct distance is the hypotenuse of the triangle formed by these coordinates: √(15² + 8²) = 17.

Adım Adım Çözüm

Plot the first movement on a coordinate plane.

Position 1 is at

(35, 0)

Setting the starting base camp at the origin

(0, 0)

, moving East means traveling along the positive x-axis.

Calculate the vector and position for the second movement.

Position 2 is at

(15, 20)

Facing East (0°), a left turn of 135° results in facing North-West. Moving

20\sqrt{2}\text{ m}

diagonally at a 45° angle to the axes translates to

\Delta x = -20\text{ m}

(West) and

\Delta y = +20\text{ m}

(North). New coordinate:

(35 - 20, 0 + 20)

Calculate the vector and position for the third movement.

Position 3 is at

(15, 8)

Facing North-West, another left turn of 135° places her facing directly South. Walking

12\text{ m}

South subtracts 12 from the y-coordinate. New coordinate:

(15, 20 - 12)

Calculate the shortest straight-line distance to the origin.

The final distance is

17\text{ m}

The straight-line distance from

(0, 0)

(15, 8)

forms a right-angled triangle. Using Pythagoras:

d = \sqrt{15^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17

Anahtar Kavram

Calculating continuous displacement using angular turns, vector breakdown, and the Pythagorean theorem.

Tahmini Süre:1m 30s

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