High tides occur approximately 12.5 hours apart. Suppose that a harbor is 50 feet deep at high tide and 30 feet deep at low tide. Find a function that models the water depth t hours after high tide. [t = 0 is a high tide]
High tides occur approximately 12.5 hours apart. Suppose that a harbor is 50 feet deep at high tide and 30 feet deep at low tide. Find a function that models the water depth t hours after high tide. [t = 0 is a high tide]
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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![**Problem 2: Modeling Tides in a Harbor**
High tides occur approximately 12.5 hours apart. Suppose that a harbor is 50 feet deep at high tide and 30 feet deep at low tide. Find a function that models the water depth \( t \) hours after high tide. [ \( t = 0 \) is a high tide]
**Explanation:**
To model the water depth in terms of time \( t \), we can use a sinusoidal function because the tides follow a consistent, wave-like pattern. The following steps outline how to derive this function:
1. **Amplitude:** The amplitude is half the difference between the high tide and low tide depths.
\[
\text{Amplitude} = \frac{50 - 30}{2} = 10 \text{ feet}
\]
2. **Midline:** The midline is the average of the high and low tide depths.
\[
\text{Midline} = \frac{50 + 30}{2} = 40 \text{ feet}
\]
3. **Period:** The period of the function is 12.5 hours because high tides occur every 12.5 hours.
\[
\text{Period} = 12.5 \text{ hours}
\]
4. **Frequency:** The frequency is the reciprocal of the period.
\[
\text{Frequency} = \frac{1}{12.5}
\]
5. **Function:** A cosine function is often used for modeling tides, as it starts from the maximum value at \( t = 0 \). The function can be written as:
\[
D(t) = 40 + 10 \cos\left(\frac{2\pi}{12.5}t\right)
\]
Where:
- \( D(t) \) represents the depth of the water \( t \) hours after high tide.
- \( 40 \) is the midline, representing the average depth.
- \( 10 \) is the amplitude.
- \( \cos\left(\frac{2\pi}{12.5}t\right) \) represents the cosine wave over the period of 12.5 hours.
This function provides a mathematical model for predicting the harbor's water depth at any given time after high tide.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff08af0c6-4dde-44ec-9e21-243b109290d6%2F5644ee1b-4f35-4bb5-acc7-5ef02f66020d%2F1fqt64_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 2: Modeling Tides in a Harbor**
High tides occur approximately 12.5 hours apart. Suppose that a harbor is 50 feet deep at high tide and 30 feet deep at low tide. Find a function that models the water depth \( t \) hours after high tide. [ \( t = 0 \) is a high tide]
**Explanation:**
To model the water depth in terms of time \( t \), we can use a sinusoidal function because the tides follow a consistent, wave-like pattern. The following steps outline how to derive this function:
1. **Amplitude:** The amplitude is half the difference between the high tide and low tide depths.
\[
\text{Amplitude} = \frac{50 - 30}{2} = 10 \text{ feet}
\]
2. **Midline:** The midline is the average of the high and low tide depths.
\[
\text{Midline} = \frac{50 + 30}{2} = 40 \text{ feet}
\]
3. **Period:** The period of the function is 12.5 hours because high tides occur every 12.5 hours.
\[
\text{Period} = 12.5 \text{ hours}
\]
4. **Frequency:** The frequency is the reciprocal of the period.
\[
\text{Frequency} = \frac{1}{12.5}
\]
5. **Function:** A cosine function is often used for modeling tides, as it starts from the maximum value at \( t = 0 \). The function can be written as:
\[
D(t) = 40 + 10 \cos\left(\frac{2\pi}{12.5}t\right)
\]
Where:
- \( D(t) \) represents the depth of the water \( t \) hours after high tide.
- \( 40 \) is the midline, representing the average depth.
- \( 10 \) is the amplitude.
- \( \cos\left(\frac{2\pi}{12.5}t\right) \) represents the cosine wave over the period of 12.5 hours.
This function provides a mathematical model for predicting the harbor's water depth at any given time after high tide.
Expert Solution
Step 1
Given,
High tide=50 feet
Low tide=30 feet
t=0 is high tide occurs at 12.5 hours
Period=24 (one complete day)
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