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Chapter 2: Problem 66

A ship sails east for 7 miles and then south for 3 miles. How far is the ship from its starting point?

Short Answer

Expert verified
The ship is \(\sqrt{58}\) miles away from its starting point (approximately 7.62 miles).

Step by step solution

01

Identify the given distances and directions

The ship sails east for 7 miles and south for 3 miles. These distances form a right-angled triangle, where the eastward distance is one leg, the southward distance is the other leg, and the distance from the starting point is the hypotenuse.
02

Apply the Pythagorean theorem

The Pythagorean theorem states that, in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In this case, let's denote the distance from the starting point (the hypotenuse) as D, the eastward distance as A (7 miles), and the southward distance as B (3 miles). Then the Pythagorean theorem can be written as: \(D^2 = A^2 + B^2\)
03

Substitute the given distances into the equation

Substitute the values of A and B into the equation: \(D^2 = (7)^2 + (3)^2\)
04

Simplify the equation

Simplify the equation by computing the squares of 7 and 3: \(D^2 = 49 + 9\)
05

Solve for the distance D

Combine the values on the right side of the equation and then take the square root of both sides to find the value of D: \(D^2 = 58\) \(D = \sqrt{58}\)
06

Express the final result

The ship is \(\sqrt{58}\) miles away from its starting point (approximately 7.62 miles).

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