After you set $100 = 68 + 122e^{-0.07t}$, what can you do first to get $e^{-0.07t}$ alone on one side? Think about subtracting and then dividing.

Once you have an equation of the form $e^{f(t)} = \text{number}$, what operation lets you bring that exponent down so you can solve for $t$?

After you get a formula for $t$ involving a logarithm and a division by $0.07$, enter it into your calculator in one line to avoid rounding errors, then round your final result to the nearest whole minute.

In Desmos, type y = 68 + 122*e^(-0.07x) to graph the temperature $T$ as a function of time $x$ (in minutes).

On a new line, type y = 100 to draw a horizontal line representing a temperature of $100^\circ\text{F}$.

Use Desmos to click on the point where the curve $y = 68 + 122e^{-0.07x}$ intersects the line $y = 100$. Read the $x$-coordinate of this intersection; that $x$-value is the time in minutes when the rod reaches $100^\circ\text{F}$. Round that $x$-value to the nearest whole number.

We are told that

and we want the time when the temperature is $100^\circ\text{F}$. So set $T(t)=100$:

Now we will solve this equation for $t$.

First subtract 68 from both sides:

Next divide both sides by 122:

Now the exponential part $e^{-0.07t}$ is isolated.

To solve for $t$, take the natural log of both sides. Using the fact that $\ln(e^x)=x$:

Now solve this linear equation for $t$ by dividing both sides by $-0.07$:

This expression gives the time (in minutes).

Use a calculator to evaluate the expression:

• Compute $32/122 \approx 0.2623$.
• Then find $\ln(0.2623) \approx -1.34$ (a negative number, as expected for a number less than 1).
• Divide by $-0.07$:

Rounding $19.1$ to the nearest whole minute gives $t \approx 19$ minutes, so the correct choice is B) 19.