La temperatura en la pista de hielo debe permanecer por debajo de [tex]50^{\circ} F[/tex]. Esta mañana la temperatura era de [tex]71^{\circ} F[/tex]. La pista de hielo cuenta con un dispositivo de refrigeración que puede reducir la temperatura [tex]3.5^{\circ} C[/tex] cada hora. ¿Qué desigualdad muestra cuánto tiempo tardará la temperatura en bajar de [tex]50^{\circ} F[/tex]?

A) [tex]50 + 3.5x \ < \ 71[/tex]

B) [tex]71 - 3.5x \ < \ 50[/tex]

C) [tex]71 - 3.5x \leq 50[/tex]

D) [tex]71 + 3.5x \ < \ 50[/tex]

We start with an initial temperature of \[tex]$71^{\circ}F\$[/tex], and we want the temperature to become less than \[tex]$50^{\circ}F\$[/tex]. The cooling device reduces the temperature by \[tex]$3.5^{\circ}F\$[/tex] per hour. If we let \[tex]$x\$[/tex] be the number of hours, the temperature after \[tex]$x\$[/tex] hours is given by

[tex]$$
T = 71 - 3.5x.
$$[/tex]

We need this temperature to be less than \[tex]$50^{\circ}F\$[/tex], so we set up the inequality:

[tex]$$
71 - 3.5x < 50.
$$[/tex]

This inequality shows the condition on \[tex]$x\$[/tex] for the temperature to drop below \[tex]$50^{\circ}F\$[/tex].

To understand the cooling process further, we can solve the inequality step by step:

1. Subtract 71 from both sides:

[tex]$$
71 - 3.5x - 71 < 50 - 71 \quad \Longrightarrow \quad -3.5x < -21.
$$[/tex]

2. Divide both sides by [tex]$-3.5$[/tex] (remembering to reverse the inequality sign when dividing by a negative number):

[tex]$$
x > \frac{-21}{-3.5}.
$$[/tex]

Simplifying the fraction:

[tex]$$
x > 6.
$$[/tex]

This tells us that it will take more than 6 hours for the temperature to drop below \[tex]$50^{\circ}F\$[/tex].

Thus, the correct inequality that represents the situation is

[tex]$$
\boxed{71 - 3.5x < 50},
$$[/tex]

which corresponds to option B.