Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Transcription and Explanation of the Image for Educational Website**
**Title: Calculation of Distance Using the Pythagorean Theorem**
**Description:**
In this diagram, we observe a typical application of the Pythagorean Theorem in a real-world scenario. The diagram showcases a right-angled triangle with the following elements:
1. **Point L:** This is located on an island, represented by a green and red circular area.
2. **Point P:** This is located on a shoreline, marked by various rocks and sandy terrain.
**Measurements:**
- The distance from Point P to a point directly below Point L on the shoreline is 4 miles.
- The distance from Point P to Point L (the hypotenuse of the right triangle) is 8 miles.
The side of the triangle running vertically from Point L straight down to the shoreline forms the height of the right-angled triangle, and this height can be calculated using the Pythagorean Theorem.
**Explanation of the Pythagorean Theorem:**
The Pythagorean Theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Symbolically, this can be written as:
\[ a^2 + b^2 = c^2 \]
Where:
- \(c\) is the hypotenuse.
- \(a\) and \(b\) are the lengths of the other two sides.
In this scenario:
- \( c = 8 \) miles (the hypotenuse),
- \( b = 4 \) miles (the base, the horizontal distance from P to the point directly under L on the shoreline),
- \( a \) is the unknown side (the vertical distance from the shoreline directly under L to L, i.e., the height).
Using the Pythagorean Theorem:
\[ a^2 + 4^2 = 8^2 \]
\[ a^2 + 16 = 64 \]
\[ a^2 = 64 - 16 \]
\[ a^2 = 48 \]
\[ a = \sqrt{48} \]
\[ a = 4\sqrt{3} \approx 6.93 \text{ miles} \]
**Conclusion:**
By applying the Pythagorean Theorem, we can calculate the height \( a \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd7c95f9d-3b81-4486-b124-731ba8663b30%2F2988dddd-d3fc-4af0-a479-fc8d8f9f1929%2F9hvu8c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Transcription and Explanation of the Image for Educational Website**
**Title: Calculation of Distance Using the Pythagorean Theorem**
**Description:**
In this diagram, we observe a typical application of the Pythagorean Theorem in a real-world scenario. The diagram showcases a right-angled triangle with the following elements:
1. **Point L:** This is located on an island, represented by a green and red circular area.
2. **Point P:** This is located on a shoreline, marked by various rocks and sandy terrain.
**Measurements:**
- The distance from Point P to a point directly below Point L on the shoreline is 4 miles.
- The distance from Point P to Point L (the hypotenuse of the right triangle) is 8 miles.
The side of the triangle running vertically from Point L straight down to the shoreline forms the height of the right-angled triangle, and this height can be calculated using the Pythagorean Theorem.
**Explanation of the Pythagorean Theorem:**
The Pythagorean Theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Symbolically, this can be written as:
\[ a^2 + b^2 = c^2 \]
Where:
- \(c\) is the hypotenuse.
- \(a\) and \(b\) are the lengths of the other two sides.
In this scenario:
- \( c = 8 \) miles (the hypotenuse),
- \( b = 4 \) miles (the base, the horizontal distance from P to the point directly under L on the shoreline),
- \( a \) is the unknown side (the vertical distance from the shoreline directly under L to L, i.e., the height).
Using the Pythagorean Theorem:
\[ a^2 + 4^2 = 8^2 \]
\[ a^2 + 16 = 64 \]
\[ a^2 = 64 - 16 \]
\[ a^2 = 48 \]
\[ a = \sqrt{48} \]
\[ a = 4\sqrt{3} \approx 6.93 \text{ miles} \]
**Conclusion:**
By applying the Pythagorean Theorem, we can calculate the height \( a \
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