**Problem Statement:** 4. Let the function \( v(t) = \frac{6 \sqrt[3]{t}}{4t + 1} \) be the volume, in cm\(^3\), of a balloon at time \( t \), in seconds. At what rate is the volume of the balloon changing with respect to time when \( t = 8 \)? Is the balloon inflating or deflating at this time? **Explanation:** - The function \( v(t) \) represents the volume of the balloon in cubic centimeters as a function of time \( t \) in seconds. - To determine the rate at which the volume is changing, calculate the derivative of \( v(t) \) with respect to \( t \), denoted as \( v'(t) \). - Evaluate \( v'(t) \) at \( t = 8 \). - The sign of \( v'(8) \) will indicate whether the balloon is inflating (positive rate) or deflating (negative rate) at \( t = 8 \).
**Problem Statement:** 4. Let the function \( v(t) = \frac{6 \sqrt[3]{t}}{4t + 1} \) be the volume, in cm\(^3\), of a balloon at time \( t \), in seconds. At what rate is the volume of the balloon changing with respect to time when \( t = 8 \)? Is the balloon inflating or deflating at this time? **Explanation:** - The function \( v(t) \) represents the volume of the balloon in cubic centimeters as a function of time \( t \) in seconds. - To determine the rate at which the volume is changing, calculate the derivative of \( v(t) \) with respect to \( t \), denoted as \( v'(t) \). - Evaluate \( v'(t) \) at \( t = 8 \). - The sign of \( v'(8) \) will indicate whether the balloon is inflating (positive rate) or deflating (negative rate) at \( t = 8 \).
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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![**Problem Statement:**
4. Let the function \( v(t) = \frac{6 \sqrt[3]{t}}{4t + 1} \) be the volume, in cm\(^3\), of a balloon at time \( t \), in seconds. At what rate is the volume of the balloon changing with respect to time when \( t = 8 \)? Is the balloon inflating or deflating at this time?
**Explanation:**
- The function \( v(t) \) represents the volume of the balloon in cubic centimeters as a function of time \( t \) in seconds.
- To determine the rate at which the volume is changing, calculate the derivative of \( v(t) \) with respect to \( t \), denoted as \( v'(t) \).
- Evaluate \( v'(t) \) at \( t = 8 \).
- The sign of \( v'(8) \) will indicate whether the balloon is inflating (positive rate) or deflating (negative rate) at \( t = 8 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F569db0f8-be64-498b-98dc-01e49b6c2a48%2F314b5c64-847c-4a12-bebe-81394597b169%2F3xyw8ut.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
4. Let the function \( v(t) = \frac{6 \sqrt[3]{t}}{4t + 1} \) be the volume, in cm\(^3\), of a balloon at time \( t \), in seconds. At what rate is the volume of the balloon changing with respect to time when \( t = 8 \)? Is the balloon inflating or deflating at this time?
**Explanation:**
- The function \( v(t) \) represents the volume of the balloon in cubic centimeters as a function of time \( t \) in seconds.
- To determine the rate at which the volume is changing, calculate the derivative of \( v(t) \) with respect to \( t \), denoted as \( v'(t) \).
- Evaluate \( v'(t) \) at \( t = 8 \).
- The sign of \( v'(8) \) will indicate whether the balloon is inflating (positive rate) or deflating (negative rate) at \( t = 8 \).
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