Answered

The average daily temperature, [tex]t[/tex], in degrees Fahrenheit for a city as a function of the month of the year, [tex]m[/tex], can be modeled by the equation

[tex]t = 35 \cos \left(\frac{\pi}{6}(m+3)\right) + 55[/tex],

where [tex]m = 0[/tex] represents January 1, [tex]m = 1[/tex] represents February 1, [tex]m = 2[/tex] represents March 1, and so on.

Which equation also models this situation?

A. [tex]t = -35 \sin \left(\frac{\pi}{6} m\right) + 55[/tex]
B. [tex]t = -35 \sin \left(\frac{\pi}{6}(m+3)\right) + 55[/tex]
C. [tex]t = 35 \sin \left(\frac{\pi}{6} m\right) + 55[/tex]
D. [tex]t = 35 \sin \left(\frac{\pi}{6}(m+3)\right) + 55[/tex]



Answer :

GinnyAnswer
To determine which equation also models the average daily temperature, we start by analyzing the given equation for temperature [tex]\( t \)[/tex]:

[tex]\[ t = 35 \cos \left( \frac{\pi}{6}(m+3) \right) + 55 \][/tex]

We will examine each of the given options to see if any of them are equivalent to the above equation.

Option 1: [tex]\( t = -35 \sin \left( \frac{\pi}{6} m \right) + 55 \)[/tex]

Rewrite the given equation in a different form using trigonometric identities. We need to express it in terms of sine, knowing that cosine can be converted to sine using the identities:

[tex]\[ \cos x = \sin \left( \frac{\pi}{2} - x \right) \][/tex]

So,

[tex]\[ t = 35 \cos \left( \frac{\pi}{6}(m+3) \right) + 55 \][/tex]

Using the trigonometric identity:

[tex]\[ \cos \left( \frac{\pi}{6}(m+3) \right) = \sin \left( \frac{\pi}{2} - \frac{\pi}{6}(m+3) \right) \][/tex]

Simplify the argument:

[tex]\[ \cos \left( \frac{\pi}{6}(m+3) \right) = \sin \left( \frac{\pi}{2} - \left( \frac{\pi}{6}(m+3) \right) \right) = \sin \left( \frac{\pi}{2} - \frac{\pi}{6}m - \frac{3\pi}{6} \right) = \sin \left( \frac{\pi}{2} - \frac{\pi}{6}m - \frac{\pi}{2} \right) = \sin \left( -\frac{\pi}{6}m \right) \][/tex]

We know:

[tex]\[ \sin(-x) = -\sin(x) \][/tex]

Thus:

[tex]\[ \sin \left( -\frac{\pi}{6}m \right) = -\sin \left( \frac{\pi}{6}m \right) \][/tex]

We then rewrite the equation as:

[tex]\[ t = 35 \cdot (-\sin \left( \frac{\pi}{6}m \right)) + 55 = -35 \sin \left( \frac{\pi}{6}m \right) + 55 \][/tex]

This matches Option 1 perfectly.

Hence, the equation that also models the average daily temperature is:

[tex]\[ t = -35 \sin \left( \frac{\pi}{6} m \right) + 55 \][/tex]

So, the correct answer is:

[tex]\[ t = -35 \sin \left( \frac{\pi}{6} m \right) + 55 \][/tex]

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