A delivery drone departs from a central launch pad and travels due north for 12 miles, the… — ACT

A delivery drone departs from a central launch pad and travels due north for $12$ miles, then turns and travels due east for $16$ miles to reach Drop Point P. A secondary relay station is located $18$ miles due south of the central launch pad. What is the straight-line distance, in miles, from Drop Point P to the secondary relay station?

A
$20$
B
$28$
C
$30$
$34$ Answer
E
$46$

Answer

The straight-line distance from Drop Point P to the secondary relay station is

34

miles.

To find the straight-line distance from Drop Point P to the secondary relay station, model the displacement as a right triangle. The horizontal distance is

16

miles (east). The vertical distance is the sum of

12

miles north and

18

miles south, giving

30

miles. Using the Pythagorean Theorem,

d = \sqrt{16^2 + 30^2} = \sqrt{256 + 900} = \sqrt{1156} = 34

miles.

Step-by-Step Solution

Set up a coordinate grid relative to the central launch pad.

Central launch pad is at

(0, 0)

. Drop Point P is at

(16, 12)

. Secondary relay station is at

(0, -18)

Establishing coordinates converts the movement into horizontal and vertical components.

Calculate the horizontal and vertical distances between Drop Point P and the secondary relay station.

Horizontal distance =

|16 - 0| = 16

miles. Vertical distance =

|12 - (-18)| = 12 + 18 = 30

miles.

These distances represent the two perpendicular legs of a right triangle.

Apply the Pythagorean Theorem to find the hypotenuse.

Distance

= \\sqrt{16^2 + 30^2} = \\sqrt{256 + 900} = \\sqrt{1156} = 34

miles.

The straight-line distance is the hypotenuse of the right triangle formed by the horizontal and vertical legs.

Key Concept

Pythagorean Theorem for distance in perpendicular directions

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