Suppose that a velocity function is given by v left parenthesis t right parenthesis equals 7 t Superscript 4 Baseline .
 
Find the position function​ s(t) if s left parenthesis 0 right parenthesis equals 8.
Question content area bottom
Part 1
A.s left parenthesis t right parenthesis equals 28 t cubed plus 8
s left parenthesis t right parenthesis equals 28 t cubed plus 8
B.s left parenthesis t right parenthesis equals seven fifths t Superscript 5
s left parenthesis t right parenthesis equals seven fifths t Superscript 5
C.s left parenthesis t right parenthesis equals 7 t Superscript 5 Baseline plus 8
s left parenthesis t right parenthesis equals 7 t Superscript 5 Baseline plus 8
D.s left parenthesis t right parenthesis equals seven fifths t Superscript 5 Baseline plus 8
s left parenthesis t right parenthesis equals seven fifths t Superscript 5 Baseline plus 8

To find the position function \( s(t) \), we need to integrate the velocity function \( v(t) = 7t^4 \), and then use the given condition \( s(0) = 8 \) to determine the constant of integration.

### Step 1: Integrate the velocity function
\[
v(t) = 7t^4
\]
The indefinite integral is:
\[
\int 7t^4 \, dt = \frac{7}{5}t^5 + C
\]
where \( C \) is the constant of integration.

### Step 2: Solve for \( C \) using \( s(0) = 8 \)
The position function is:
\[
s(t) = \frac{7}{5}t^5 + C
\]
Substitute \( t = 0 \) and \( s(0) = 8 \):
\[
s(0) = \frac{7}{5}(0)^5 + C = 8
\]
\[
C = 8
\]

### Step 3: Write the position function
\[
s(t) = \frac{7}{5}t^5 + 8
\]

### Final Answer:
The correct choice is:
\[
\text{D. } s(t) = \frac{7}{5}t^5 + 8
\]

---

Would you like further clarification or a breakdown of any specific step?  

**Here are 5 related questions to explore further:**  
1. What is the physical meaning of the constant of integration in position functions?  
2. How would you find acceleration from the given velocity function?  
3. If the velocity function were \( v(t) = 7t^3 \), how would the position function change?  
4. How does the initial condition \( s(0) \) affect the solution?  
5. What is the difference between definite and indefinite integration?  

**Tip:** Always remember to apply initial conditions after integrating to solve for the constant of integration!

Suppose that a velocity function is given by v(t) = 7t^4. Find the position function s(t) if s(0) = 8. Which of the following is correct?
A. s(t) = 28t^3 + 8
B. s(t) = (7/5)t^5
C. s(t) = 7t^5 + 8
D. s(t) = (7/5)t^5 + 8

Learn how to calculate the position function s(t) by integrating the velocity function v(t) = 7t^4 and applying the initial condition s(0) = 8. This calculus problem demonstrates key concepts of integration and the Fundamental Theorem of Calculus, ideal for Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation