Interpret the exponential function word problem

Full solution

Q. At the moment a hot iron rod is plunged into freezing water, the difference between the rod's and the water's temperatures is

100^{\circ}

Celsius. This causes the iron to cool and the temperature difference drops by

60 \%

every second.

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Write a function that gives the temperature difference in degrees Celsius,

D(t), t

seconds after the rod was plunged into the water.

\newline

D(t)=\square

Identify Initial Temperature Difference: Step $1$ : Identify the initial temperature difference when the iron rod is plunged into the water. The problem states that the initial temperature difference is $100$ degrees Celsius. This is the value of $D(0)$ , the temperature difference at time $t = 0$ seconds.
Determine Rate of Decrease: Step $2$ : Determine the rate at which the temperature difference decreases. The problem states that the temperature difference drops by $60\%$ every second. This means that each second, the temperature difference is $40\%$ ( $100\% - 60\%$ ) of what it was the previous second.
Express Rate as Decimal: Step $3$ : Express the rate of decrease as a decimal to be used in the exponential decay function. The rate of $40\%$ as a decimal is $0.40$ .
Write Exponential Decay Function: Step $4$ : Write the exponential decay function. The general form of an exponential decay function is $D(t) = D(0) \times (\text{decay factor})^{t}$ , where $D(0)$ is the initial amount, decay factor is the percentage of the amount remaining after each time interval, and $t$ is the time in seconds.
Substitute Known Values: Step $5$ : Substitute the known values into the exponential decay function. The initial temperature difference is $D(0) = 100$ degrees Celsius, and the decay factor is $0.40$ . Therefore, the function is $D(t) = 100 \times 0.40^{t}$ .
Simplify Function: Step $6$ : Simplify the function if necessary. In this case, the function is already in its simplest form, so no further simplification is needed.