Anticipating Exponential Growth \[ \begin{array}{l}\text { At } 11: 30 \mathrm{a.m} \text {. the bottle is } 1 / 4 \text { of the way full. At what } \\ \text { time will the bottle be } 1 / 2 \text { full? } \\ \text { O } 11: 31 \mathrm{a} . \mathrm{m} . \\ 11: 35 \mathrm{a} . \mathrm{m} . \\ 011: 40 \mathrm{a.m} . \\ 1: 00 \mathrm{p.m} .\end{array} \]

When the amount of liquid in a bottle is growing exponentially, the key thing to remember is that it doesn't take long to reach the next halfway point. If the bottle is \( \frac{1}{4} \) full at 11:30 a.m., it will double to \( \frac{1}{2} \) full just a short time later. In fact, for exponential growth, if something is \( \frac{1}{4} \) full, then it will take only one more time interval to reach \( \frac{1}{2} \) full. So, since the bottle is \( \frac{1}{4} \) full at 11:30 a.m., and assuming a steady growth rate, it will be \( \frac{1}{2} \) full by 11:31 a.m. It’s like watching your favorite plant perk up with a little sunlight – growth can be quite the surprise!