Question

1. a. in city a, the temperature at 6 a.m. is represented by variable x. write an expression to represent the temperature (represented by variable t) at noon.

b. the temperature at noon was twice the temperature at 6 a.m. write an algebraic expression that describes the relationship between the temperature t at noon and the temperature at 6 a.m.
c. the temperature at 1 p.m. was 2 °c higher than the temperature at noon. the temperature at 1 p.m. was 0 °c, determine the temperature at 6 a.m.

Explanation:

Part b

Step1: Define variables

Let \( x \) be the temperature at 6 a.m. and \( t \) be the temperature at noon.

Step2: Relate the temperatures

Given that the temperature at noon (\( t \)) is twice the temperature at 6 a.m. (\( x \)), we can write the relationship as \( t = 2x \).

Step1: Relate 1 p.m. and noon

Let \( t_1 \) be 1 p.m. temp, \( t \) noon, \( x \) 6 a.m. temp. Given \( t_1 = t + 2 \).

Step2: Substitute \( t_1 \) value

If \( t_1 = 0^\circ \text{C} \), then \( 0 = t + 2 \implies t = -2^\circ \text{C} \).

Step3: Use part b relation

From part b, \( t = 2x \), so \( -2 = 2x \implies x = \frac{-2}{2} = -1^\circ \text{C} \).

Answer:

\( t = 2x \)

Part c (assuming the temperature at 1 p.m. is related to noon, and we need to find 6 a.m. temperature. Let's assume the temperature at 1 p.m. is \( t_1 \), noon is \( t \), 6 a.m. is \( x \))

First, from part b: \( t = 2x \)
Given that the temperature at 1 p.m. (\( t_1 \)) is \( 2^\circ \text{C} \) higher than noon (\( t \)), so \( t_1 = t + 2 \)
Also, assume the temperature at 1 p.m. was given (maybe from context, but let's suppose we know \( t_1 \), but if we assume the temperature at 1 p.m. was, say, let's check the original problem again. Wait, the original problem for part c: "The temperature at 1 p.m. was \( 2^\circ \text{C} \) higher than the temperature at noon. The temperature at 1 p.m. was \( 0^\circ \text{C} \), determine the temperature at 6 a.m." Wait, maybe a typo, but let's re-express:

Let's correct: Suppose "The temperature at 1 p.m. was \( 2^\circ \text{C} \) higher than the temperature at noon. The temperature at 1 p.m. was \( 0^\circ \text{C} \)? No, that might not make sense. Wait, maybe "The temperature at 1 p.m. was \( 2^\circ \text{C} \) higher than the temperature at noon. The temperature at 6 a.m. was \( 0^\circ \text{C} \), determine the temperature at noon and 1 p.m." No, the original text: "c. The temperature at 1 p.m. was \( 2^\circ \text{C} \) higher than the temperature at noon. The temperature at 1 p.m. was \( 0^\circ \text{C} \), determine the temperature at 6 a.m." Wait, maybe a mistake, but let's proceed with the given:

From part b: \( t = 2x \) (noon \( t \), 6 a.m. \( x \))
Temperature at 1 p.m. (\( t_1 \)) = \( t + 2 \)
If \( t_1 = 0^\circ \text{C} \), then \( t + 2 = 0 \implies t = -2^\circ \text{C} \)
Then from \( t = 2x \), \( -2 = 2x \implies x = -1^\circ \text{C} \)