A worker record the temperature T°C at various time on a certain winter day. He found that the temperature was 5°C, 8°C and 7°C at 2 hours, 4 hours and 6 hours after 9 am respectively. Find an expression for T at time T hours after 9 am.
Find an expression
You cannot use T for both time and temperature!
since the variation is not linear, you will need to approximate.
It dropped one degree in the last 2 hours, so at that same rate,
temp = 7 - 1/3 T
Actually, I suspect a typo.
To find an expression for the temperature at time T hours after 9 am, we need to analyze the given information and make a general formula.
Let's break down the problem step by step:
1. The worker recorded the temperature at various times - 2 hours, 4 hours, and 6 hours after 9 am.
2. The temperature recorded was 5°C, 8°C, and 7°C at those respective times.
3. We need to find a general expression for the temperature at time T hours after 9 am.
To derive the expression, we can observe a pattern in the given data:
- The temperature changes from 5°C to 8°C over a span of 2 hours and then from 8°C to 7°C over a span of 2 hours as well.
Based on this pattern, we can conclude that the temperature changes linearly. This means that the temperature increases by 3°C in the first 2-hour period and then decreases by 1°C in the next 2-hour period.
Now, to find the expression for the temperature at time T hours after 9 am, we need to determine the relationship between T and the temperature. Since the temperature changes linearly, we can use a linear equation.
Let's break it down into two cases:
1. If T is within the 2-hour interval (0 ≤ T < 2):
- The initial temperature is 5°C, and it increases by 3°C every 2 hours.
- So, for T hours within this interval, the temperature can be expressed as: T°C.
2. If T is within the next 2-hour interval (2 ≤ T < 4):
- The temperature at the end of the previous interval is 5°C + 3°C = 8°C.
- The temperature then decreases by 1°C every 2 hours.
- So, for T hours within this interval, the temperature can be expressed as: 8 - (T - 2)°C.
Now, we can write an expression for the temperature at time T hours after 9 am:
- If 0 ≤ T < 2: T°C
- If 2 ≤ T < 4: 8 - (T - 2)°C
- If T ≥ 4: 7°C
This expression covers the temperature variation for different intervals of T hours after 9 am based on the given data.