Suppose the equation below models the height h(t) of a distress flare signal at time t seconds. Assume the interval under consideration is [0, 49] where 49 seconds is the time the flare fizzles out and hits the ground. h (t) = -16t² +656t+6272 The Mean Value Theorem guarantees the existence of some value t from 0 to 49 seconds when the instantaneous velocity of the flare is equal to the average velocity. Find this value.
Suppose the equation below models the height h(t) of a distress flare signal at time t seconds. Assume the interval under consideration is [0, 49] where 49 seconds is the time the flare fizzles out and hits the ground. h (t) = -16t² +656t+6272 The Mean Value Theorem guarantees the existence of some value t from 0 to 49 seconds when the instantaneous velocity of the flare is equal to the average velocity. Find this value.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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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: Application of the Mean Value Theorem in Kinematics**
Suppose the equation below models the height \( h(t) \) of a distress flare signal at time \( t \) seconds. Assume the interval under consideration is \([0, 49]\) where 49 seconds is the time the flare fizzles out and hits the ground.
\[
h(t) = -16t^2 + 656t + 6272
\]
The Mean Value Theorem guarantees the existence of some value \( t \) from 0 to 49 seconds when the instantaneous velocity of the flare is equal to the average velocity. Find this value.
**Analysis:**
- **Equation Explanation**: The quadratic equation \( h(t) = -16t^2 + 656t + 6272 \) represents the height of the flare over time.
- **Interval**: The model considers the time from 0 to 49 seconds, with 49 seconds marking the end of the flare’s flight.
**Concept Application**:
- **Mean Value Theorem**: This theorem states that for a continuous function over an interval \([a, b]\), there exists at least one point \( c \) in the interval where the instantaneous rate of change (derivative) is equal to the average rate of change over \([a, b]\).
To apply the theorem, find:
1. The average velocity of the flare over the interval \([0, 49]\).
2. The instantaneous velocity as given by the derivative \( h'(t) \).
3. Solve for the time \( t \) when these two velocities are equal.
**Graphs or Diagrams**: None in the provided content. The focus is on understanding the equation and application of the Mean Value Theorem in kinematics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1fe61787-87ec-42ef-9b74-8862c323e0b5%2F51759c26-2b7a-453d-b17a-fd84295d732b%2Fb01kd0s_processed.png&w=3840&q=75)
Transcribed Image Text:**Topic: Application of the Mean Value Theorem in Kinematics**
Suppose the equation below models the height \( h(t) \) of a distress flare signal at time \( t \) seconds. Assume the interval under consideration is \([0, 49]\) where 49 seconds is the time the flare fizzles out and hits the ground.
\[
h(t) = -16t^2 + 656t + 6272
\]
The Mean Value Theorem guarantees the existence of some value \( t \) from 0 to 49 seconds when the instantaneous velocity of the flare is equal to the average velocity. Find this value.
**Analysis:**
- **Equation Explanation**: The quadratic equation \( h(t) = -16t^2 + 656t + 6272 \) represents the height of the flare over time.
- **Interval**: The model considers the time from 0 to 49 seconds, with 49 seconds marking the end of the flare’s flight.
**Concept Application**:
- **Mean Value Theorem**: This theorem states that for a continuous function over an interval \([a, b]\), there exists at least one point \( c \) in the interval where the instantaneous rate of change (derivative) is equal to the average rate of change over \([a, b]\).
To apply the theorem, find:
1. The average velocity of the flare over the interval \([0, 49]\).
2. The instantaneous velocity as given by the derivative \( h'(t) \).
3. Solve for the time \( t \) when these two velocities are equal.
**Graphs or Diagrams**: None in the provided content. The focus is on understanding the equation and application of the Mean Value Theorem in kinematics.
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