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Chapter 6: Problem 78

The flow rate (or water discharge rate) at the mouth of the Orinoco River in South America may be approximated by $$ F(t)=26,000 \sin \left[\frac{\pi}{6}(t-5.5)\right]+34,000 $$ where \(t\) is the time in months and \(F(t)\) is the flow rate in \(\mathrm{m}^{3} / \mathrm{sec} .\) For approximately how many months each year does the flow rate exceed \(55,000 \mathrm{m}^{3} / \mathrm{sec} ?\)

Short Answer

Expert verified
The flow rate exceeds 55,000 m鲁/s for about 5 months per year.

Step by step solution

01

Understand the Flow Rate Function

We are given the flow rate function \( F(t) = 26,000 \sin\left[\frac{\pi}{6}(t-5.5)\right] + 34,000 \). This function models the flow rate in cubic meters per second depending on the time of year, where \( t \) is in months.
02

Set Up the Inequality

We need to find when \( F(t) > 55,000 \). Substitute \( F(t) \) in the inequality: \[ 26,000 \sin\left[\frac{\pi}{6}(t-5.5)\right] + 34,000 > 55,000 \] This simplifies to: \[ 26,000 \sin\left[\frac{\pi}{6}(t-5.5)\right] > 21,000 \] Divide both sides by 26,000: \[ \sin\left[\frac{\pi}{6}(t-5.5)\right] > \frac{21,000}{26,000} \] Simplify the fraction: \[ \sin\left[\frac{\pi}{6}(t-5.5)\right] > \frac{21}{26} \approx 0.8077 \]
03

Determine When the Sine Condition Holds

The inequality \( \sin(x) > 0.8077 \) implies \( x \) is in the interval \( \left[ \frac{5\pi}{18}, \frac{13\pi}{18} \right] \) plus any integer multiple of \( 2\pi \). Solving for the principal value:\[ \frac{\pi}{6}(t-5.5) = x \] Implies \[ x = \frac{\pi}{6}(t-5.5) \]
04

Solve for t in Terms of Given Inequality

Rewrite the sine condition bounds for \( x \):\[ \frac{\pi}{6}(t-5.5) = \frac{5\pi}{18} \quad \text{to} \quad \frac{13\pi}{18} \] Solving for \( t \) when:1. \( \frac{\pi}{6}(t-5.5) = \frac{5\pi}{18} \) - Multiply both sides by \( \frac{6}{\pi} \) - \( t - 5.5 = \frac{5}{3} \) - \( t = \frac{5}{3} + 5.5 = 7.1667 \approx 7.17 \)2. \( \frac{\pi}{6}(t-5.5) = \frac{13\pi}{18} \) - Multiply both sides by \( \frac{6}{\pi} \) - \( t - 5.5 = \frac{13}{3} \) - \( t = \frac{13}{3} + 5.5 = 9.8333 \approx 9.83 \)
05

Calculate the Duration Each Year

Since the sine function is periodic with a period of 12 months, this condition will occur twice a year for approximately 2.67 months each time. Therefore, the total time for which the flow rate exceeds 55,000 m鲁/s is approximately 5.34 months.
06

Round Off to Nearest Month

Since 5.34 months approximates to about 5 months most closely, we conclude that the flow rate exceeds 55,000 m鲁/s for about 5 months per year.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

sine function
The sine function, known widely for its role in trigonometry, is used to model periodic phenomena like waves, circles, and oscillations. In the flow rate function given, the sine element is at the core of modeling seasonal changes in water discharge.
  • The function given, \( F(t) = 26,000 \sin\left[\frac{\pi}{6}(t-5.5)\right] + 34,000 \), uses the sine function to reflect how the flow rate varies over months.
  • The term \( \frac{\pi}{6}(t - 5.5) \) determines the shape and shift of the sine curve, helping to correctly align it with seasonal peaks in flow.
Understanding sine functions involves recognizing how they oscillate between -1 and 1. These oscillations depict natural rhythms found in real-world data. They repeat in cycles, which in this context, spans 12 months given the annual periodicity.
trigonometric inequality
In our exercise, we set up a trigonometric inequality to determine when the flow rate is greater than a specific threshold. This involves analyzing the equation \( F(t) > 55,000 \) to find when it holds true.
  • By reformulating the equation to \( 26,000 \sin\left[\frac{\pi}{6}(t-5.5)\right] > 21,000 \), we focus solely on the sine component.
  • Dividing by 26,000 yields \( \sin\left[\frac{\pi}{6}(t-5.5)\right] > 0.8077 \).
Solving such inequalities means identifying which parts of the sine wave fall above the specified value, utilizing properties of sine's periodicity. In this example, the sine values greater than 0.8077 occur within specific arcs or segments of its cycle, dictating the months exceeding the flow threshold.
periodic functions
Periodic functions repeat their values in regular intervals, over a specified period. In this exercise, the function \( F(t) = 26,000 \sin\left[\frac{\pi}{6}(t-5.5)\right] + 34,000 \) is periodic, signifying seasonal oscillation in river discharge.
  • The sine function's period is calculated from its coefficient: \( \frac{\pi}{6} \) corresponds to a 12-month cycle, aligning with annual changes.
  • A period of 12 months means the pattern of flow rate repeats every year. Thus, high flow rates recur twice each year, governed by the sine wave's cycle.
Periodic functions efficiently model phenomena that have natural repeats or cycles in their behavior over timeframes such as days, months, or years, making them indispensable in various fields including hydrology and ecology.
mathematical modeling
Mathematical modeling involves creating equations to represent real-world situations. In our example, the equation \( F(t) = 26,000 \sin\left[\frac{\pi}{6}(t-5.5)\right] + 34,000 \) serves as a model for the Orinoco River's changing flow rate.
  • This model helps predict river discharge at any given month \( t \), based on historical patterns.
  • Mathematical models, like this one, often inform decisions in water management, predicting seasonal variations for agriculture, and understanding ecological impacts.
The key to successful modeling is appropriately choosing functions that capture the essence of what is being studied. Through modeling, complex systems become simplified representations that illuminate trends, predict future occurrences, and aid in strategic planning across scientific and engineering fields.

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