A gauge at the end of a pier measures and tracks the water depth d (in feet) over time. Regression analysis was performed to fit a trigonometric function to the data. The function d(t) models the depth of the water over time, d(t) 11 sin(0.406t) + 21 where t represents the number of hours past midnight and 0 ≤ t ≤ 24. High tide occurs twice in a day. After how many hours will the second high tide occur? O 3.87 hours O 15. 48 hours O 19.35 hours 23. 21 hours
A gauge at the end of a pier measures and tracks the water depth d (in feet) over time. Regression analysis was performed to fit a trigonometric function to the data. The function d(t) models the depth of the water over time, d(t) 11 sin(0.406t) + 21 where t represents the number of hours past midnight and 0 ≤ t ≤ 24. High tide occurs twice in a day. After how many hours will the second high tide occur? O 3.87 hours O 15. 48 hours O 19.35 hours 23. 21 hours
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.2: Graphs Of Equations
Problem 45E
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Question
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### Analyzing Tidal Patterns Using Trigonometric Functions
A gauge at the end of a pier measures and tracks the water depth \( d \) (in feet) over time. Regression analysis was performed to fit a trigonometric function to the data.
The function \( d(t) \) models the depth of the water over time,
\[
d(t) = 11 \sin(0.406t) + 21
\]
where \( t \) represents the number of hours past midnight and \( 0 \leq t \leq 24 \).
**Problem:**
High tide occurs twice in a day. After how many hours will the second high tide occur?
**Options:**
- [ ] 3.87 hours
- [ ] 15.48 hours
- [ ] 19.35 hours
- [ ] 23.21 hours
The correct choice is **highlighted**:
- [x] **3.87 hours**
---
This function and question are a typical example of how trigonometric functions can be used to model periodic natural phenomena, such as tide patterns. The function describes a sine wave with an amplitude of 11 feet and a vertical shift of 21 feet, indicating the average water depth. Therefore, the water depth oscillates between 10 feet (21-11) and 32 feet (21+11) over the 24-hour period. The coefficient of \( t \) inside the sine function determines the period of the oscillations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf693007-feb9-4e65-ba86-b1b16e800487%2Fd873159d-779d-460b-bbf8-e9a410458c30%2F3dfglf_processed.png&w=3840&q=75)
Transcribed Image Text:---
### Analyzing Tidal Patterns Using Trigonometric Functions
A gauge at the end of a pier measures and tracks the water depth \( d \) (in feet) over time. Regression analysis was performed to fit a trigonometric function to the data.
The function \( d(t) \) models the depth of the water over time,
\[
d(t) = 11 \sin(0.406t) + 21
\]
where \( t \) represents the number of hours past midnight and \( 0 \leq t \leq 24 \).
**Problem:**
High tide occurs twice in a day. After how many hours will the second high tide occur?
**Options:**
- [ ] 3.87 hours
- [ ] 15.48 hours
- [ ] 19.35 hours
- [ ] 23.21 hours
The correct choice is **highlighted**:
- [x] **3.87 hours**
---
This function and question are a typical example of how trigonometric functions can be used to model periodic natural phenomena, such as tide patterns. The function describes a sine wave with an amplitude of 11 feet and a vertical shift of 21 feet, indicating the average water depth. Therefore, the water depth oscillates between 10 feet (21-11) and 32 feet (21+11) over the 24-hour period. The coefficient of \( t \) inside the sine function determines the period of the oscillations.
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