Sure, let's work through the problem step-by-step.
1. Define the initial conditions and rate of change:
- The initial temperature is [tex]\(60^{\circ} F\)[/tex].
- The temperature decreases by [tex]\(3^{\circ} F\)[/tex] each hour.
- Let [tex]\(h\)[/tex] be the number of hours that have passed.
2. Set up the inequality:
- We need to find when the temperature will be below [tex]\(32^{\circ} F\)[/tex].
- The formula for the temperature after [tex]\(h\)[/tex] hours of decrease is:
[tex]\[
\text{Temperature after } h \text{ hours} = 60 - 3h
\][/tex]
- We want this temperature to be below [tex]\(32^{\circ} F\)[/tex]:
[tex]\[
60 - 3h < 32
\][/tex]
3. Solve the inequality:
- Start with the inequality:
[tex]\[
60 - 3h < 32
\][/tex]
- Subtract 60 from both sides to isolate the term involving [tex]\(h\)[/tex]:
[tex]\[
-3h < 32 - 60
\][/tex]
[tex]\[
-3h < -28
\][/tex]
- Divide both sides by [tex]\(-3\)[/tex]. When dividing by a negative number, remember to reverse the inequality sign:
[tex]\[
h > \frac{-28}{-3}
\][/tex]
[tex]\[
h > \frac{28}{3}
\][/tex]
- Perform the division:
[tex]\[
h > 9.33
\][/tex]
So, the inequality that represents the time when the temperature will drop below [tex]\(32^{\circ} F\)[/tex] is:
[tex]\[
60 - 3h < 32
\][/tex]
The correct answer from the options provided is:
D. [tex]\(60 - 3h < 32\)[/tex]
