Consider a map oriented so that the x-axis runs east-west (with east being the "positive" direction) and y runs north-south (with north "positive"). A person drives 28 km to the north, turns and drives 73 km to the east, and then turns north and drives for an unknown distance z. If his final position is 97 km from where he started, find z. km
Consider a map oriented so that the x-axis runs east-west (with east being the "positive" direction) and y runs north-south (with north "positive"). A person drives 28 km to the north, turns and drives 73 km to the east, and then turns north and drives for an unknown distance z. If his final position is 97 km from where he started, find z. km
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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![**Problem Description:**
Consider a map oriented so that the x-axis runs east-west (with east being the "positive" direction) and the y-axis runs north-south (with north "positive"). A person drives 28 km to the north, turns and drives 73 km to the east, and then turns north and drives for an unknown distance \( z \). If his final position is 97 km from where he started, find \( z \).
**Solution:**
Let’s represent the movement in a coordinate system. Starting from the origin:
1. Move 28 km north: The coordinates are (0,28).
2. Move 73 km east: The coordinates become (73, 28).
3. Move \( z \) km north: The coordinates are (73, 28 + \( z \)).
We use the distance formula to find \( z \):
Given the final distance from the start to end is 97 km, we have:
\[
\sqrt{(73 - 0)^2 + ((28 + z) - 0)^2} = 97
\]
Simplifying,
\[
\sqrt{73^2 + (28 + z)^2} = 97
\]
Squaring both sides,
\[
73^2 + (28 + z)^2 = 97^2
\]
Calculating,
\( 5329 + 784 + 56z + z^2 = 9409 \)
Rearrange terms to form a quadratic equation:
\( z^2 + 56z + 6113 = 9409 \)
\( z^2 + 56z - 3296 = 0 \)
Solving this quadratic equation results in the value of \( z \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F31367789-01e4-4129-b30b-49084de51bdb%2F44d1c3f6-6dd3-4845-a0b7-17e745a6a11d%2Fxn9k4ob_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Description:**
Consider a map oriented so that the x-axis runs east-west (with east being the "positive" direction) and the y-axis runs north-south (with north "positive"). A person drives 28 km to the north, turns and drives 73 km to the east, and then turns north and drives for an unknown distance \( z \). If his final position is 97 km from where he started, find \( z \).
**Solution:**
Let’s represent the movement in a coordinate system. Starting from the origin:
1. Move 28 km north: The coordinates are (0,28).
2. Move 73 km east: The coordinates become (73, 28).
3. Move \( z \) km north: The coordinates are (73, 28 + \( z \)).
We use the distance formula to find \( z \):
Given the final distance from the start to end is 97 km, we have:
\[
\sqrt{(73 - 0)^2 + ((28 + z) - 0)^2} = 97
\]
Simplifying,
\[
\sqrt{73^2 + (28 + z)^2} = 97
\]
Squaring both sides,
\[
73^2 + (28 + z)^2 = 97^2
\]
Calculating,
\( 5329 + 784 + 56z + z^2 = 9409 \)
Rearrange terms to form a quadratic equation:
\( z^2 + 56z + 6113 = 9409 \)
\( z^2 + 56z - 3296 = 0 \)
Solving this quadratic equation results in the value of \( z \).
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