m(t₂)−m(t₁)=15(t₂ − t₁ )
A chemist dissolves sodium acetate in boiling water. As the temperature cools down, the crystals start to grow with increasing mass m(t), in grams (g) t seconds (s) after crystallization. The equation above shows this relationship where t₁ < t₂. Which of the following correctly explains the growth of the crystal mass?
The crystal mass grows linearly by 15g per s
The crystal mass grows exponentially by 1g every 15s
The crystal mass grows linearly by a factor 15 every s
For me, the best way to think about this one is that it’s a translation question and a plug-in question. Here’s what I mean.
The part on the left side of the equation, , is basically how much bigger the mass is after the longer time than it is after the shorter time. So (here comes the plugging in part—it’s just to help us conceptualize, not necessarily calculate) if is 5 seconds and is 8 seconds, then is the difference between the mass at 8 seconds and the mass at 5 seconds.
On the right side of the equation, we have a 15, and then more ‘s. To make it a bit less abstract, let’s keep using 5 and 8 for now, which will make the right side look like .
So now on the left we have an expression that tells us the difference in crystal mass between 8 seconds and 5 seconds, and on the right, we have 15 times that same time interval. The crystal is 45 grams bigger after 8 seconds of growth than it was after 5 seconds of growth. It grew by 15 grams per second.
Your choices are cut off here (I don’t see a D) but it doesn’t matter because the right one is in there: we want choice A.
Another way to get this one is by eliminating. The formula you’re given isn’t exactly simple, but even a very quick glance confirms it doesn’t have any exponents, so you can cross off B. C uses the word linearly, but it also says it grows by a factor of 15, which implies exponential growth. That’s just a mess—cross it off!