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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Question
What is the domain of the function
![**Problem Statement:**
Consider a setup where a child, 4 feet tall, is standing some distance \( d \) feet away from a street lamp. The street lamp has a height of 12 feet. We are interested in finding the length of the shadow cast by the light shining on the child, which we will denote as \( l \).
**Diagram Explanation:**
The provided diagram illustrates the scenario with the following elements:
1. **Street Lamp:** Positioned vertically and denoted as a 12 feet tall structure.
2. **Child:** Standing to the left of the street lamp, with a height of 4 feet.
3. **Distance \( d \):** The horizontal distance between the child and the base of the street lamp.
4. **Shadow \( l \):** The length of the shadow cast by the child on the ground, extending from the child's feet to the tip of the shadow.
The diagram includes visual cues:
- A right-angle triangle formed by the height of the street lamp and the ground.
- Another smaller right-angle triangle formed by the child's height, the ground, and the shadow.
- An extending light ray from the top of the street lamp, passing over the child's head and hitting the ground at the end of the shadow.
**Mathematical Model:**
To solve for the length of the shadow \( l \) given the height of the child and street lamp, we use similar triangles. The key relationships can be derived as follows:
Since the two triangles are similar:
\[ \frac{\text{Child's Height}}{\text{Shadow Length}} = \frac{\text{Lamp's Height}}{\text{Distance} + \text{Shadow Length}} \]
Hence, the lengths need to satisfy the equation:
\[ \frac{4}{l} = \frac{12}{d+l} \]
This equation can be solved to find \( l \) in terms of \( d \).
---
This setup is useful for understanding principles of geometric similarity and proportionality, particularly in applied contexts like shadow calculations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe91e659-0274-4af0-b893-ea272a52ff02%2F8ce7e7a3-7f91-48cc-93f2-4e49198d083b%2Fm3kmps7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Consider a setup where a child, 4 feet tall, is standing some distance \( d \) feet away from a street lamp. The street lamp has a height of 12 feet. We are interested in finding the length of the shadow cast by the light shining on the child, which we will denote as \( l \).
**Diagram Explanation:**
The provided diagram illustrates the scenario with the following elements:
1. **Street Lamp:** Positioned vertically and denoted as a 12 feet tall structure.
2. **Child:** Standing to the left of the street lamp, with a height of 4 feet.
3. **Distance \( d \):** The horizontal distance between the child and the base of the street lamp.
4. **Shadow \( l \):** The length of the shadow cast by the child on the ground, extending from the child's feet to the tip of the shadow.
The diagram includes visual cues:
- A right-angle triangle formed by the height of the street lamp and the ground.
- Another smaller right-angle triangle formed by the child's height, the ground, and the shadow.
- An extending light ray from the top of the street lamp, passing over the child's head and hitting the ground at the end of the shadow.
**Mathematical Model:**
To solve for the length of the shadow \( l \) given the height of the child and street lamp, we use similar triangles. The key relationships can be derived as follows:
Since the two triangles are similar:
\[ \frac{\text{Child's Height}}{\text{Shadow Length}} = \frac{\text{Lamp's Height}}{\text{Distance} + \text{Shadow Length}} \]
Hence, the lengths need to satisfy the equation:
\[ \frac{4}{l} = \frac{12}{d+l} \]
This equation can be solved to find \( l \) in terms of \( d \).
---
This setup is useful for understanding principles of geometric similarity and proportionality, particularly in applied contexts like shadow calculations.
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