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The temperature at $12$ noon was $10^{\circ}\ C$ above zero. If it decreases at the rate of $2^{\circ}\ C$ per hour until midnight, at what time would the temperature be $8^{\circ}\ C$ below zero? What would be the temperature at mid-night?
Given:
The temperature at $12$ noon was $10^{\circ}\ C$ above zero. It decreases at the rate of $2^{\circ}\ C$ per hour until midnight.
To do:
We have to find the time at which temperature will be $8^{\circ}\ C$ and the temperature at midnight.
Solution:
Initial temperature $=10^{\circ}\ C$
Rate of decrease in temperature per hour $=2^{\circ}\ C$
The above situation can be written in an equation form as,
Temperature $(T) = (10-2x)^{\circ}\ C$ where $x$ is the time between noon and midnight.
The time at which the temperature would be $8^{\circ}\ C$ below zero is,
$-8 = (10-2x)$
$-8 = 10-2x$
$2x=10+8$
$2x=18$
$x=\frac{18}{2}$
$x=9$
Therefore, at $9\ PM$ the temperature would be $8^{\circ}\ C$ below zero.
The temperature at mid-night($12\ AM$) $=$ The decrease in temperature between 12 noon($12\ PM$) and midnight($12\ AM$)
The decrease in temperature in $12$ hours $= 12 \times 2^{\circ}\ C$
$= 24^{\circ}\ C$
The temperature at midnight $=10^{\circ}\ C-24^{\circ}\ C$
$=-14^{\circ}\ C$
Therefore, the temperature at midnight would be $14^{\circ}\ C$ below zero.
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