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The regression equation [tex]y = 8.6 \sin (0.24 x - 1.88) + 62.8[/tex] models the Fahrenheit temperature at midnight. What is the maximum temperature in the room during the first 24 hours?

A. [tex]54.2^{\circ} F[/tex]
B. [tex]57.0^{\circ} F[/tex]
C. [tex]62.8^{\circ} F[/tex]
D. [tex]71.4^{\circ} F[/tex]

Answer :

We are given the regression equation

[tex]$$
y = 8.6 \sin (0.24x - 1.88) + 62.8.
$$[/tex]

This equation represents the Fahrenheit temperature, where the sine part is multiplied by an amplitude of [tex]$8.6$[/tex] and then shifted vertically by [tex]$62.8$[/tex]. The sine function has a maximum value of [tex]$1$[/tex]. Therefore, the maximum value of the entire function occurs when

[tex]$$
\sin(0.24x - 1.88) = 1.
$$[/tex]

At this point, the temperature is

[tex]$$
y_{\text{max}} = 8.6 \times 1 + 62.8 = 8.6 + 62.8 = 71.4.
$$[/tex]

Thus, the maximum temperature in the room during the first 24 hours is

[tex]$$
71.4^{\circ}F.
$$[/tex]

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