A triangle is called equilateral if the length of its three sides are the same. The surface are | and the perimeter P of an equilateral triangle of a given length s for each side are provided, V3 A = 4 P = 3s and he perimeter of an equilateral triangle is increasing at a rate of 5 feet per hour. Find the correspondin ate of change of its area when the perimeter is 6 feet. Include units.
A triangle is called equilateral if the length of its three sides are the same. The surface are | and the perimeter P of an equilateral triangle of a given length s for each side are provided, V3 A = 4 P = 3s and he perimeter of an equilateral triangle is increasing at a rate of 5 feet per hour. Find the correspondin ate of change of its area when the perimeter is 6 feet. Include units.
Calculus: Early Transcendentals
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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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![### Topic: Equilateral Triangles - Area and Perimeter Relationships
#### Definition:
A triangle is called equilateral if the length of its three sides are the same.
#### Key Formulas:
The surface area \( A \) and the perimeter \( P \) of an equilateral triangle of a given length \( s \) for each side are provided as follows:
\[
A = \frac{\sqrt{3}}{4} s^2
\]
\[
P = 3s
\]
#### Problem Statement:
The perimeter of an equilateral triangle is increasing at a rate of 5 feet per hour. Find the corresponding rate of change of its area when the perimeter is 6 feet. Include units.
**Explanation and Solution Approach:**
1. **Perimeter-Related Details:**
- Given perimeter rate of change: \( \frac{dP}{dt} = 5 \) feet per hour.
- Perimeter equation: \( P = 3s \).
2. **Calculate Side Length (s):**
- When \( P = 6 \) feet:
\[
6 = 3s \implies s = 2 \text{ feet}
\]
3. **Derive the Rate of Change of Side Length:**
- Differentiate \( P = 3s \) with respect to time \( t \):
\[
\frac{dP}{dt} = 3 \frac{ds}{dt}
\]
- Substitute \( \frac{dP}{dt} = 5 \) feet/hour:
\[
5 = 3 \frac{ds}{dt} \implies \frac{ds}{dt} = \frac{5}{3} \text{ feet per hour}
\]
4. **Area-Related Rates:**
- Area equation: \(\ A = \frac{\sqrt{3}}{4} s^2 \).
- Differentiate with respect to \( t \):
\[
\frac{dA}{dt} = \frac{\sqrt{3}}{4} \cdot 2s \cdot \frac{ds}{dt} = \frac{\sqrt{3}}{2} s \cdot \frac{ds}{dt}
\]
- Substitute \( s = 2 \) feet and \( \frac{ds](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2c26efb-761a-4d98-b451-c4611947b84e%2Fcc66b8e9-6575-4808-9ca4-62c9f8f32938%2Fwrecfqu_processed.png&w=3840&q=75)
Transcribed Image Text:### Topic: Equilateral Triangles - Area and Perimeter Relationships
#### Definition:
A triangle is called equilateral if the length of its three sides are the same.
#### Key Formulas:
The surface area \( A \) and the perimeter \( P \) of an equilateral triangle of a given length \( s \) for each side are provided as follows:
\[
A = \frac{\sqrt{3}}{4} s^2
\]
\[
P = 3s
\]
#### Problem Statement:
The perimeter of an equilateral triangle is increasing at a rate of 5 feet per hour. Find the corresponding rate of change of its area when the perimeter is 6 feet. Include units.
**Explanation and Solution Approach:**
1. **Perimeter-Related Details:**
- Given perimeter rate of change: \( \frac{dP}{dt} = 5 \) feet per hour.
- Perimeter equation: \( P = 3s \).
2. **Calculate Side Length (s):**
- When \( P = 6 \) feet:
\[
6 = 3s \implies s = 2 \text{ feet}
\]
3. **Derive the Rate of Change of Side Length:**
- Differentiate \( P = 3s \) with respect to time \( t \):
\[
\frac{dP}{dt} = 3 \frac{ds}{dt}
\]
- Substitute \( \frac{dP}{dt} = 5 \) feet/hour:
\[
5 = 3 \frac{ds}{dt} \implies \frac{ds}{dt} = \frac{5}{3} \text{ feet per hour}
\]
4. **Area-Related Rates:**
- Area equation: \(\ A = \frac{\sqrt{3}}{4} s^2 \).
- Differentiate with respect to \( t \):
\[
\frac{dA}{dt} = \frac{\sqrt{3}}{4} \cdot 2s \cdot \frac{ds}{dt} = \frac{\sqrt{3}}{2} s \cdot \frac{ds}{dt}
\]
- Substitute \( s = 2 \) feet and \( \frac{ds
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