Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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![### Problem Statement
Given the vector function \(\vec{r}(t) = \langle t, e^t, 10 \rangle\), find the tangential component of acceleration \(a_T\) when \(t = 7\).
---
### Solution
To find the tangential component of acceleration \(a_T\), we follow these steps:
1. **Compute the velocity vector \(\vec{v}(t)\):**
\[
\vec{v}(t) = \frac{d\vec{r}}{dt} = \left\langle \frac{d}{dt}(t), \frac{d}{dt}(e^t), \frac{d}{dt}(10) \right\rangle = \langle 1, e^t, 0 \rangle
\]
2. **Compute the acceleration vector \(\vec{a}(t)\):**
\[
\vec{a}(t) = \frac{d\vec{v}}{dt} = \left\langle \frac{d}{dt}(1), \frac{d}{dt}(e^t), \frac{d}{dt}(0) \right\rangle = \langle 0, e^t, 0 \rangle
\]
3. **Find the magnitude of the velocity vector \(|\vec{v}(t)|\):**
\[
|\vec{v}(t)| = \sqrt{ (1)^2 + (e^t)^2 + (0)^2 } = \sqrt{1 + e^{2t}}
\]
4. **Calculate the tangential component of acceleration \(a_T\):**
\[
a_T = \frac{\vec{v}(t) \cdot \vec{a}(t)}{|\vec{v}(t)|}
\]
Here, \(\vec{v}(t) \cdot \vec{a}(t)\) is the dot product of the velocity and acceleration vectors:
\[
\vec{v}(t) \cdot \vec{a}(t) = \langle 1, e^t, 0 \rangle \cdot \langle 0, e^t, 0 \rangle = 1 \cdot 0 + e^t \cdot](https://content.bartleby.com/qna-images/question/e4962ae7-1cb2-4276-8f6a-b9f851b67289/e1320306-c25b-41ae-a139-057598576334/1oq7465_processed.png)
Transcribed Image Text:### Problem Statement
Given the vector function \(\vec{r}(t) = \langle t, e^t, 10 \rangle\), find the tangential component of acceleration \(a_T\) when \(t = 7\).
---
### Solution
To find the tangential component of acceleration \(a_T\), we follow these steps:
1. **Compute the velocity vector \(\vec{v}(t)\):**
\[
\vec{v}(t) = \frac{d\vec{r}}{dt} = \left\langle \frac{d}{dt}(t), \frac{d}{dt}(e^t), \frac{d}{dt}(10) \right\rangle = \langle 1, e^t, 0 \rangle
\]
2. **Compute the acceleration vector \(\vec{a}(t)\):**
\[
\vec{a}(t) = \frac{d\vec{v}}{dt} = \left\langle \frac{d}{dt}(1), \frac{d}{dt}(e^t), \frac{d}{dt}(0) \right\rangle = \langle 0, e^t, 0 \rangle
\]
3. **Find the magnitude of the velocity vector \(|\vec{v}(t)|\):**
\[
|\vec{v}(t)| = \sqrt{ (1)^2 + (e^t)^2 + (0)^2 } = \sqrt{1 + e^{2t}}
\]
4. **Calculate the tangential component of acceleration \(a_T\):**
\[
a_T = \frac{\vec{v}(t) \cdot \vec{a}(t)}{|\vec{v}(t)|}
\]
Here, \(\vec{v}(t) \cdot \vec{a}(t)\) is the dot product of the velocity and acceleration vectors:
\[
\vec{v}(t) \cdot \vec{a}(t) = \langle 1, e^t, 0 \rangle \cdot \langle 0, e^t, 0 \rangle = 1 \cdot 0 + e^t \cdot
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