On Friday, the temperature at 8:00 a.m. Was `49^(@)F` and rose at a constant rate of `1/2^(@)F` per hour until noon. A cold front passed through at noon, and the temperature then fell at a constant rete of `1^(@)F` per hour. The temperature first fell below `49^(@)F` between:

Options

A. noon and 1 p.m.
B. 1 p.m. And 2 p.m.
C. 2 p.m. And 3 p.m.
D. 3 p.m. And 4 p.m.

Answer

To solve the problem step by step, we will analyze the temperature changes throughout the day.

### Step 1: Determine the temperature at noon
The temperature at 8:00 a.m. is given as \(49^\circ F\). The temperature rises at a constant rate of \( \frac{1}{2}^\circ F\) per hour until noon.

The time from 8:00 a.m. to noon is 4 hours.

To find the temperature at noon, we can calculate:
\[ \text{Temperature at noon} = \text{Initial temperature} + \left(\text{Rate of increase} \times \text{Time}\right) \]
\[ \text{Temperature at noon} = 49 + \left(\frac{1}{2} \times 4\right) = 49 + 2 = 51^\circ F \]

### Step 2: Determine the temperature drop after noon
After noon, the temperature falls at a constant rate of \(1^\circ F\) per hour.

Let \(X\) be the number of hours after noon. The temperature at time \(X\) hours after noon can be expressed as:
\[ \text{Temperature} = 51 - 1 \times X \]

### Step 3: Set up the inequality for when the temperature falls below \(49^\circ F\)
We want to find when the temperature first falls below \(49^\circ F\). Therefore, we set up the inequality:
\[ 51 - X < 49 \]

### Step 4: Solve the inequality
Rearranging the inequality gives:
\[ 51 - 49 < X \]
\[ 2 < X \]
This means \(X\) must be greater than \(2\).

### Step 5: Determine the time when the temperature falls below \(49^\circ F\)
Since \(X\) is the number of hours after noon, and we found that \(X > 2\), this indicates that the temperature will fall below \(49^\circ F\) after 2 hours from noon.

Calculating the time:
\[ \text{Time} = 12:00 \text{ p.m.} + 2 \text{ hours} = 2:00 \text{ p.m.} \]

Thus, the temperature first falls below \(49^\circ F\) between \(2:00 \text{ p.m.}\) and \(3:00 \text{ p.m.}\).

### Final Answer
The temperature first fell below \(49^\circ F\) between \(2:00 \text{ p.m.}\) and \(3:00 \text{ p.m.}\).

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