To write a function that accurately represents the temperature difference between the cake and the cooler over time, we need to consider the following information:
1. The initial temperature difference is [tex]\( 50^{\circ} \)[/tex] Celsius.
2. The temperature difference loses [tex]\( \frac{1}{5} \)[/tex] of its value every minute.
Given these points, we can model the temperature difference as a function of time [tex]\( t \)[/tex].
Let [tex]\( D(t) \)[/tex] represent the temperature difference at time [tex]\( t \)[/tex] minutes after the cake is put in the cooler. Here's how we can derive this function:
1. Initial Value: At [tex]\( t = 0 \)[/tex], the temperature difference [tex]\( D(0) \)[/tex] is [tex]\( 50^{\circ} \)[/tex] Celsius.
2. Rate of Change: The temperature difference decreases by [tex]\( \frac{1}{5} \)[/tex] of its value every minute. This means that [tex]\( \frac{4}{5} \)[/tex] of the temperature difference remains after each minute.
We can express this reduction mathematically as:
[tex]\[ D(t) = 50 \times \left(\frac{4}{5}\right)^t \][/tex]
Here's a step-by-step explanation:
- At [tex]\( t = 0 \)[/tex]:
[tex]\[
D(0) = 50
\][/tex]
- At [tex]\( t = 1 \)[/tex]:
[tex]\[
D(1) = 50 \times \left(\frac{4}{5}\right)
\][/tex]
- At [tex]\( t = 2 \)[/tex]:
[tex]\[
D(2) = 50 \times \left(\frac{4}{5}\right) \times \left(\frac{4}{5}\right) = 50 \times \left(\frac{4}{5}\right)^2
\][/tex]
- Generally, at time [tex]\( t \)[/tex]:
[tex]\[
D(t) = 50 \times \left(\frac{4}{5}\right)^t
\][/tex]
Thus, the function that describes the temperature difference [tex]\( D(t) \)[/tex], [tex]\( t \)[/tex] minutes after the cake was put in the cooler, is given by:
[tex]\[
D(t) = 50 \times \left(\frac{4}{5}\right)^t
\][/tex]
This exponential decay function models how the temperature difference between the cake and the cooler decreases over time.
