Number 6 Certainly! Here's a transcription suitable for an educational website.---**Exploration of Velocity and Distance****5. Graph Analysis:**Below is a graph that shows the velocity, $ v $, of an object in meters per second.- **Graph Details:** - Vertical Axis: Velocity $ v $ in meters per second (m/s), ranging from 10 to 40. - Horizontal Axis: Time in seconds (t), ranging from 0 to 6. - The graph is a straight line increasing from 10 m/s at $ t = 0 $ to 40 m/s at $ t = 6 $.**Task:**Estimate the total distance the object traveled between $ t = 0 $ and $ t = 6 $.**6. Velocity Function Analysis:**The velocity of a car is given by the function:\[ v(t) = t^2 + 2t \]where $ v(t) $ is measured in feet per second (ft/sec) and $ t $ is measured in seconds.**Tasks:**a) Estimate the distance traveled from $ t = 1 $ to $ t = 4 $ using 3 subdivisions.b) Assess the accuracy of your estimation.**7. Tabulated Speed Analysis:**A car is moving along a straight road and its speed is continually increasing. Speed readings are recorded at two-second intervals and tabulated as follows:- **Time (sec):** 0, 2, 4, 6, 8- **Velocity (ft/sec):** 30, 36, 38, 40, 44**Tasks:**a) For the first two-second interval, determine the minimum and maximum distance the car could have traveled.b) Make a lower and upper estimate of the distance the car could have traveled for each two-second interval.c) If estimating the distance traveled in the eight-second interval, what would be your guess? What is the maximum possible difference between your guess and the actual distance?d) Consider if speedometer readings were available for each second. By how much would the upper estimate exceed the lower estimate?---This structured guide assists learners in understanding how to analyze velocity and distance using graphs, functions, and data tables.

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Description: Number 6 Certainly! Here's a transcription suitable for an educational website.---**Exploration of Velocity and Distance****5. Graph Analysis:**Below is a graph that shows the velocity, $ v $, of an object in meters per second.- **Graph Details:**

Answered: Number 6

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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Number 6

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**Exploration of Velocity and Distance**

**5. Graph Analysis:**
Below is a graph that shows the velocity, $ v $, of an object in meters per second.

- **Graph Details:**
- Vertical Axis: Velocity $ v $ in meters per second (m/s), ranging from 10 to 40.
- Horizontal Axis: Time in seconds (t), ranging from 0 to 6.
- The graph is a straight line increasing from 10 m/s at $ t = 0 $ to 40 m/s at $ t = 6 $.

**Task:**
Estimate the total distance the object traveled between $ t = 0 $ and $ t = 6 $.

**6. Velocity Function Analysis:**

The velocity of a car is given by the function:

\[ v(t) = t^2 + 2t \]

where $ v(t) $ is measured in feet per second (ft/sec) and $ t $ is measured in seconds.

**Tasks:**
a) Estimate the distance traveled from $ t = 1 $ to $ t = 4 $ using 3 subdivisions.

b) Assess the accuracy of your estimation.

**7. Tabulated Speed Analysis:**

A car is moving along a straight road and its speed is continually increasing. Speed readings are recorded at two-second intervals and tabulated as follows:

- **Time (sec):** 0, 2, 4, 6, 8
- **Velocity (ft/sec):** 30, 36, 38, 40, 44

**Tasks:**

a) For the first two-second interval, determine the minimum and maximum distance the car could have traveled.

b) Make a lower and upper estimate of the distance the car could have traveled for each two-second interval.

c) If estimating the distance traveled in the eight-second interval, what would be your guess? What is the maximum possible difference between your guess and the actual distance?

d) Consider if speedometer readings were available for each second. By how much would the upper estimate exceed the lower estimate?

---

This structured guide assists learners in understanding how to analyze velocity and distance using graphs, functions, and data tables.

Expert Solution

Step 1

$(6)$ The velocity of a car is given by :

$v (t) = t^{2} + 2 t$

here, $v (t)$ is measured in $f t / s e c$ and $t$ is measured in $s e c .$

$(a)$ To determine the distance travelled from $t = 1$ to $t = 4$ using $3$ subdivisions,

$\begin{array}{rcl} \int_{1}^{4} v (t) & = & \int_{1}^{2} (t^{2} + 2 t) d t + \int_{2}^{3} (t^{2} + 2 t) d t + \int_{3}^{4} (t^{2} + 2 t) d t \\ = & {(\frac{t^{3}}{3} + 2 \frac{t^{2}}{2})}_{1}^{2} + {(\frac{t^{3}}{3} + 2 \frac{t^{2}}{2})}_{2}^{3} + {(\frac{t^{3}}{3} + 2 \frac{t^{2}}{2})}_{3}^{4} \\ = & [(\frac{2^{3}}{3} + 2^{2}) - (\frac{1^{3}}{3} + 1^{2})] + [(\frac{3^{3}}{3} + 3^{2}) - (\frac{2^{3}}{3} + 2^{2})] + [(\frac{4^{3}}{3} + 4^{2}) - (\frac{3^{3}}{3} + 3^{2})] \\ = & [(\frac{8}{3} + 4) - (\frac{1}{3} + 1)] + [(9 + 9) - (\frac{8}{3} + 4)] + [(\frac{64}{3} + 16) - (9 + 9)] \\ = & [(\frac{20}{3} - \frac{4}{3})] + [(18 - \frac{20}{3})] + [(\frac{112}{3} - 18)] \\ = & \frac{16}{3} + \frac{34}{3} + \frac{58}{3} \\ = & \frac{108}{3} \\ = & 36 f t . \end{array}$

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