Question

one spring day, claire noted the time of day and the temperature, in degrees fahrenheit. her findings are as follows: at 6 a.m., the temperature was $52^{circ}$ f. for the next 6 hours, the temperature rose $2^{circ}$ per hour. for the next 3 hours, it rose $3^{circ}$ per hour. the temperature then stayed steady until 6 p.m. for the next 4 hours, the temperature dropped $2^{circ}$ per hour. the temperature then dropped steadily until the temperature was $63^{circ}$ at midnight. on the set of axes below, graph claires data.

Explanation:

To graph Claire's temperature data, we first determine the temperature at key times:

Step 1: 6 a.m. to 12 p.m. (noon)

• At 6 a.m., temperature = \( 52^\circ \text{F} \).
• For the next 6 hours (until 12 p.m.), temperature rises \( 2^\circ \) per hour.

Temperature at 12 p.m. = \( 52 + 6 \times 2 = 52 + 12 = 64^\circ \text{F} \).

Step 2: 12 p.m. to 3 p.m.

• For the next 3 hours (until 3 p.m.), temperature rises \( 3^\circ \) per hour.

Temperature at 3 p.m. = \( 64 + 3 \times 3 = 64 + 9 = 73^\circ \text{F} \).

Step 3: 3 p.m. to 6 p.m.

• Temperature stays steady until 6 p.m., so temperature at 6 p.m. = \( 73^\circ \text{F} \).

Step 4: 6 p.m. to 10 p.m.

• For the next 4 hours (until 10 p.m.), temperature drops \( 2^\circ \) per hour.

Temperature at 10 p.m. = \( 73 - 4 \times 2 = 73 - 8 = 65^\circ \text{F} \).

Step 5: 10 p.m. to midnight (12 a.m.)

• Temperature drops from \( 65^\circ \text{F} \) at 10 p.m. to \( 63^\circ \text{F} \) at midnight (2 hours later).

Rate of drop = \( \frac{65 - 63}{2} = 1^\circ \) per hour.

Key Data Points:

Time	Temperature (\(^\circ \text{F}\))
12 p.m.	64
3 p.m.	73
6 p.m.	73
10 p.m.	65
Midnight	63

To graph:

• X - axis: Time (in hours after 6 a.m., or labeled with actual times: 6 a.m., 12 p.m., 3 p.m., 6 p.m., 10 p.m., midnight).
• Y - axis: Temperature (in \(^\circ \text{F}\), from ~50 to 80).

Plot the points:

• (6 a.m., 52), (12 p.m., 64), (3 p.m., 73), (6 p.m., 73), (10 p.m., 65), (midnight, 63).

Connect the points with line segments:

• From 6 a.m. to 12 p.m.: Line with slope \( 2 \) (rising \( 2^\circ \) per hour).
• From 12 p.m. to 3 p.m.: Line with slope \( 3 \) (rising \( 3^\circ \) per hour).
• From 3 p.m. to 6 p.m.: Horizontal line (slope \( 0 \), steady temperature).
• From 6 p.m. to 10 p.m.: Line with slope \( -2 \) (dropping \( 2^\circ \) per hour).
• From 10 p.m. to midnight: Line with slope \( -1 \) (dropping \( 1^\circ \) per hour).

This graph visually represents the temperature changes throughout the day.

Answer:

To graph Claire's temperature data, we first determine the temperature at key times:

Step 1: 6 a.m. to 12 p.m. (noon)

• At 6 a.m., temperature = \( 52^\circ \text{F} \).
• For the next 6 hours (until 12 p.m.), temperature rises \( 2^\circ \) per hour.

Temperature at 12 p.m. = \( 52 + 6 \times 2 = 52 + 12 = 64^\circ \text{F} \).

Step 2: 12 p.m. to 3 p.m.

• For the next 3 hours (until 3 p.m.), temperature rises \( 3^\circ \) per hour.

Temperature at 3 p.m. = \( 64 + 3 \times 3 = 64 + 9 = 73^\circ \text{F} \).

Step 3: 3 p.m. to 6 p.m.

• Temperature stays steady until 6 p.m., so temperature at 6 p.m. = \( 73^\circ \text{F} \).

Step 4: 6 p.m. to 10 p.m.

• For the next 4 hours (until 10 p.m.), temperature drops \( 2^\circ \) per hour.

Temperature at 10 p.m. = \( 73 - 4 \times 2 = 73 - 8 = 65^\circ \text{F} \).

Step 5: 10 p.m. to midnight (12 a.m.)

• Temperature drops from \( 65^\circ \text{F} \) at 10 p.m. to \( 63^\circ \text{F} \) at midnight (2 hours later).

Rate of drop = \( \frac{65 - 63}{2} = 1^\circ \) per hour.

Key Data Points:

Time	Temperature (\(^\circ \text{F}\))
12 p.m.	64
3 p.m.	73
6 p.m.	73
10 p.m.	65
Midnight	63

To graph:

• X - axis: Time (in hours after 6 a.m., or labeled with actual times: 6 a.m., 12 p.m., 3 p.m., 6 p.m., 10 p.m., midnight).
• Y - axis: Temperature (in \(^\circ \text{F}\), from ~50 to 80).

Plot the points:

• (6 a.m., 52), (12 p.m., 64), (3 p.m., 73), (6 p.m., 73), (10 p.m., 65), (midnight, 63).

Connect the points with line segments:

• From 6 a.m. to 12 p.m.: Line with slope \( 2 \) (rising \( 2^\circ \) per hour).
• From 12 p.m. to 3 p.m.: Line with slope \( 3 \) (rising \( 3^\circ \) per hour).
• From 3 p.m. to 6 p.m.: Horizontal line (slope \( 0 \), steady temperature).
• From 6 p.m. to 10 p.m.: Line with slope \( -2 \) (dropping \( 2^\circ \) per hour).
• From 10 p.m. to midnight: Line with slope \( -1 \) (dropping \( 1^\circ \) per hour).

This graph visually represents the temperature changes throughout the day.