The temperature at 12 noon was 10°C above zero. If it decreases at the rate of 2°C per hour until midnight, at what time would the temperature be 8°C below zero? What would be the temperature at midnight?

Let's solve this problem step by step.

Given:

The temperature at 12 noon was 10°C above zero.
The temperature decreases at a rate of 2°C per hour until midnight.
We want to find out at what time the temperature would be 8°C below zero and the temperature at midnight.
Let T represent the temperature in degrees Celsius, and t represent the time in hours after 12 noon. At 12 noon, the temperature is 10°C above zero, so we can express this as:

T(0) = 10°C

The temperature decreases at a rate of 2°C per hour, so we can use the rate of change formula:

T(t) = T(0) - rate × t

Where:

T(t) is the temperature at time t hours after 12 noon.
T(0) is the initial temperature (10°C above zero).
rate is the rate of temperature decrease (2°C per hour).
t is the time in hours.
We want to find when the temperature is 8°C below zero, so we set up the equation:

T(t) = -8°C

Now, we can plug in the values and solve for t:

-8 = 10 - 2t

Subtract 10 from both sides:

-8 - 10 = -2t

-18 = -2t

Now, divide both sides by -2 to solve for t:

t = 9

So, it will take 9 hours for the temperature to be 8°C below zero, which means it will be at that temperature at 9 PM (12 noon + 9 hours).

Now, to find the temperature at midnight (12 AM), we can plug t = 12 into the formula:

T(t) = T(0) - rate × t

T(12) = 10 - 2 × 12

T(12) = 10 - 24

T(12) = -14°C

So, the temperature at midnight will be -14°C.