A cup of hot coffee has been left to cool in a room with an ambient temperature of $21^{\circ} \mathrm{C}$ . $\newline$ The relationship between the elapsed time, $m$ , in minutes, since the coffee was left to cool, and the temperature of the coffee, $T$ , measured in ${ }^{\circ} \mathrm{C}$ , is modeled by the following function. $\newline$ $T(m)=21+74 \cdot 10^{-0.03 m}$ $\newline$ What will the temperature of the coffee be after $10$ minutes? $\newline$ Round your answer, if necessary, to the nearest hundredth. $\newline$ ${ }^{\circ} C$

Full solution

Q. A cup of hot coffee has been left to cool in a room with an ambient temperature of

21^{\circ} \mathrm{C}

\newline

The relationship between the elapsed time,

m

, in minutes, since the coffee was left to cool, and the temperature of the coffee,

T

, measured in

{ }^{\circ} \mathrm{C}

, is modeled by the following function.

\newline

T(m)=21+74 \cdot 10^{-0.03 m}

\newline

What will the temperature of the coffee be after

10

minutes?

\newline

Round your answer, if necessary, to the nearest hundredth.

\newline

{ }^{\circ} C

Identify Function and Value: Identify the given function and the value to be substituted. $\newline$ The function given is $T(m) = 21 + 74 \times 10^{(-0.03m)}$ , where $m$ is the elapsed time in minutes. We need to find the temperature of the coffee after $10$ minutes.
Substitute Value into Function: Substitute the value of $m$ into the function. $\newline$ We substitute $m = 10$ into the function to find $T(10)$ . $\newline$ $T(10) = 21 + 74 \times 10^{(-0.03 \times 10)}$
Calculate Exponent: Calculate the exponent part of the function. $\newline$ Calculate $10^{(-0.03 \times 10)}$ which is $10^{(-0.3)}$ . $\newline$ $10^{(-0.3)} \approx 0.5011872336$
Multiply with $74$ : Multiply the result of the exponent with $74$ . Multiply $74$ by the result from the previous step. $74 \times 0.5011872336 \approx 37.0878608896$
Add to Find $T(10)$ : Add $21$ to the result from the previous step to find $T(10)$ . $\newline$ $T(10) = 21 + 37.0878608896$ $\newline$ $T(10) \approx 58.0878608896$
Round to Nearest Hundredth: Round the result to the nearest hundredth. $\newline$ Round $58.0878608896$ to the nearest hundredth. $\newline$ $T(10) \approx 58.09^{\circ}C$