Question

consider the table above comparing maximum and minimum daily high temperatures (\\(^{\circ}\text{f}\\)) for three different towns. suppose that the daily high temperature of these towns over time can be modeled with a sinusoidal function in the x-y coordinate plane. which description best matches the data above?

• the function modeling the daily high temperature of chi-town over time has an amplitude of 13.
• the function modeling the daily high temperature of chi-town over time has an amplitude of 50.
• the function modeling the daily high temperature of h-town over time has an amplitude of 16.
• the function modeling the daily high temperature of h-town over time has an amplitude of 79.

Explanation:

Identify the given values

Using the Mathematical Modeling knowledge point

• For Chi-Town:
• Maximum daily high temperature: \(M_{\text{Chi}} = 83^\circ\text{F}\)
• Minimum daily high temperature: \(m_{\text{Chi}} = 33^\circ\text{F}\)
• For H-Town:
• Maximum daily high temperature: \(M_{\text{H}} = 95^\circ\text{F}\)
• Minimum daily high temperature: \(m_{\text{H}} = 63^\circ\text{F}\)

Calculate the amplitudes

Using the Sinusoidal Transformations knowledge point

$$ \text{Amplitude} = \frac{\text{Maximum} - \text{Minimum}}{2} $$

• For Chi-Town:

$$ A_{\text{Chi}} = \frac{83 - 33}{2} = \frac{50}{2} = 25 $$

• For H-Town:

$$ A_{\text{H}} = \frac{95 - 63}{2} = \frac{32}{2} = 16 $$

Evaluate the options

Using the Mathematical Modeling knowledge point

• Option 1: Chi-Town amplitude is 13 (Incorrect, it is 25).
• Option 2: Chi-Town amplitude is 50 (Incorrect, 50 is the range).
• Option 3: H-Town amplitude is 16 (Correct).
• Option 4: H-Town amplitude is 79 (Incorrect, 79 is the midline).

Answer:

• The function modeling the daily high temperature of Chi-Town over time has an amplitude of 13.
• The function modeling the daily high temperature of Chi-Town over time has an amplitude of 50.
• The function modeling the daily high temperature of H-Town over time has an amplitude of 16. (Correct answer)
• The function modeling the daily high temperature of H-Town over time has an amplitude of 79.