This no-calc prompt is from the Daily Practice app:

The temp T in °C of a chilled drink after the drink has been sitting on a table for m min is represented by T(m) = 32 – 28*3^(-0.05m).

What is the best interpretation of the # 32 in this context?

A. The drink is originally 32°C.
B. Every 32 min, the temperature warms by 3°C.
C. After 32 min, the drink will fully warm to the ambient temp.
D. After the drink has been sitting for a very long time, the temp of the drink will approach 32°C.

I’m going to try to talk you through my thought process for this question rather than just do math.

First step for me here is to eliminate B and C—we know that m is minutes and that appears in the exponent in this function, far away from the 32.

From there,* your job is to figure out whether the expression will approach 32 as m increases (in which case the answer is D) or get farther away from 32 as m increases (in which case the answer is A).

Think about the equation this way: T(m)=32-\text{[something]}. Of course, that something is 28\times 3^{-0.05m}, but stay with me for a minute. If the [something] is going to grow, then A will be the answer. If the something is going to shrink, then D will be the answer.

Now, does 28\times 3^{-0.05m} grow or shrink?

We know a negative exponent is the same as the inverse of the positive exponent, so let’s look at it that way:

    \begin{align*}28\times \frac{1}{3^{0.05m}}\end{align*}

When m = 0, that’s going to just be 28:

    \begin{align*}28\times \frac{1}{3^{0.05(0)}}\\28\times \frac{1}{3^0}}\\28\times \frac{1}{1}\\28\end{align*}

When m grows to 20, however…

    \begin{align*}28\times \frac{1}{3^{0.05(20)}}\\28\times \frac{1}{3^1}}\\28\times \frac{1}{3}\\9.333...\end{align*}

When m grows, our [something] shrinks! Therefore, the function will get closer to 32 as m increases, and D is the answer!


* From the get-go, you might also be able to eliminate A if you know a few things about Celsius. 32° C is about 90° F, which is in no way a “chilled” drink. In that case, you really need not do any math at all!

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