A student drives 15 miles west from home and then 30 miles north to go to work. How much shorter would the trip be if it was possible to drive along a straight line from home to work.
A student drives 15 miles west from home and then 30 miles north to go to work. How much shorter would the trip be if it was possible to drive along a straight line from home to work.
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
![**Problem Statement:**
A student drives 15 miles west from home and then 30 miles north to go to work. How much shorter would the trip be if it was possible to drive along a straight line from home to work?
**Solution Explanation:**
To solve this problem, we can use the Pythagorean theorem to find the straight-line distance (the hypotenuse) from the student’s home to work. The student’s path forms a right triangle with legs of 15 miles and 30 miles.
Using the Pythagorean theorem:
\[
c = \sqrt{a^2 + b^2}
\]
where:
- \( a = 15 \) miles,
- \( b = 30 \) miles.
Calculating:
\[
c = \sqrt{15^2 + 30^2} = \sqrt{225 + 900} = \sqrt{1125} = 33.54 \text{ miles (approximately)}
\]
Therefore, the student travels approximately \( 15 + 30 = 45 \) miles total using the original route. The straight-line distance is approximately \( 33.54 \) miles.
**Conclusion:**
The straight-line path would save the student about \( 45 - 33.54 = 11.46 \) miles.
**Diagrams:**
In a hypothetical diagram, the path would be visualized as a right triangle with the westward path (15 miles) on the x-axis and the northward path (30 miles) on the y-axis. The hypotenuse would represent the direct path from home to work.
This example demonstrates the application of the Pythagorean theorem to find the shortest distance between two points on a coordinate plane.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ee59d88-150a-48ec-8795-2bb12b8e59c7%2Ff1f0bd12-151a-4cac-94d3-d271a08e2821%2Fj2cb6m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
A student drives 15 miles west from home and then 30 miles north to go to work. How much shorter would the trip be if it was possible to drive along a straight line from home to work?
**Solution Explanation:**
To solve this problem, we can use the Pythagorean theorem to find the straight-line distance (the hypotenuse) from the student’s home to work. The student’s path forms a right triangle with legs of 15 miles and 30 miles.
Using the Pythagorean theorem:
\[
c = \sqrt{a^2 + b^2}
\]
where:
- \( a = 15 \) miles,
- \( b = 30 \) miles.
Calculating:
\[
c = \sqrt{15^2 + 30^2} = \sqrt{225 + 900} = \sqrt{1125} = 33.54 \text{ miles (approximately)}
\]
Therefore, the student travels approximately \( 15 + 30 = 45 \) miles total using the original route. The straight-line distance is approximately \( 33.54 \) miles.
**Conclusion:**
The straight-line path would save the student about \( 45 - 33.54 = 11.46 \) miles.
**Diagrams:**
In a hypothetical diagram, the path would be visualized as a right triangle with the westward path (15 miles) on the x-axis and the northward path (30 miles) on the y-axis. The hypotenuse would represent the direct path from home to work.
This example demonstrates the application of the Pythagorean theorem to find the shortest distance between two points on a coordinate plane.
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