Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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**Estimating the Height of a Building using Free-Fall Motion**

**Problem Statement:**

To estimate the height of a building, a stone is dropped from the top of the building into a pool of water at ground level. The splash is observed 5.7 seconds after the stone is dropped. How high is the building? Use the position function for free-falling objects given below. Please round your answer to one decimal place.

**Position Function for Free-Falling Objects:**

\[ s(t) = -4.9t^2 + v_0t + s_0 \]

Where:

- \( s(t) \) is the position of the object at time \( t \)
- \( t \) is the time in seconds
- \( v_0 \) is the initial velocity (which is 0 in this case, as the stone is dropped)
- \( s_0 \) is the initial position (the height of the building in this case)

**Calculation:**

To find the height of the building (\( s_0 \)), we use the given time of 5.7 seconds and the position function:

\[ s(t) = s_0 \]

\[ 0 = -4.9(5.7)^2 + s_0 \]

\[ 0 = -158.241 + s_0 \]

\[ s_0 = 158.241 \]

Thus, the height of the building is approximately 158.2 meters. 

[**Input Box:**]
\[ \boxed{158.2} \, \text{m} \]

**Note:** Ensure to input the rounded value correctly in the provided box to get precise results.
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Transcribed Image Text:**Estimating the Height of a Building using Free-Fall Motion** **Problem Statement:** To estimate the height of a building, a stone is dropped from the top of the building into a pool of water at ground level. The splash is observed 5.7 seconds after the stone is dropped. How high is the building? Use the position function for free-falling objects given below. Please round your answer to one decimal place. **Position Function for Free-Falling Objects:** \[ s(t) = -4.9t^2 + v_0t + s_0 \] Where: - \( s(t) \) is the position of the object at time \( t \) - \( t \) is the time in seconds - \( v_0 \) is the initial velocity (which is 0 in this case, as the stone is dropped) - \( s_0 \) is the initial position (the height of the building in this case) **Calculation:** To find the height of the building (\( s_0 \)), we use the given time of 5.7 seconds and the position function: \[ s(t) = s_0 \] \[ 0 = -4.9(5.7)^2 + s_0 \] \[ 0 = -158.241 + s_0 \] \[ s_0 = 158.241 \] Thus, the height of the building is approximately 158.2 meters. [**Input Box:**] \[ \boxed{158.2} \, \text{m} \] **Note:** Ensure to input the rounded value correctly in the provided box to get precise results.
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