Assume that a certain virus being studied is currently doubling each day. Suppose that one unit of virus is contained now. Complete the table below to show the growth of the virus over time with 0 representing now, 1 representing one day from now, and \(-1\) representing one day ago.
| Number of Days | Exponential Expression (doubling means base 2) | Number of Units of Virus |
| \(-2\) (\(2\) days ago) | 1. | 2. |
| \(-1 \) (\(1\) day ago) | \(2^{-1}\) | 3. |
| \(0\) (now) | \(2^0\) | \(1\) |
| \(1\) | 4. | 5. |
| \(2\) | 6. | 7. |
| \(5\) | 8. | 9. |
Considering this pattern, let’s explore how much virus would be present in \(\frac{1}{2}\) days from now. We know that the amount of virus is between 1 and 2 units since \(\frac{1}{2}\) days is between 0 and 1 days.
- First, we can write this amount as an exponential expression with base 2 and days \(\frac{1}{2}\) as \(2^\frac{1}{2}\).
- Second, we can use the exponent multiplication rule to show that \(2^\frac{1}{2}\cdot 2^\frac{1}{2}=2^{\frac{1}{2}+\frac{1}{2}}=2^1=2 \).
- Therefore, we know that \(2^\frac{1}{2}\) is equal to \(\sqrt{2}\) since \(\sqrt{2}\cdot \sqrt{2}=2\).
- Use a calculator to determine the amount of virus present in \(\frac{1}{2}\) days from now which is the same as \(2^\frac{1}{2}\). Round the answer to the nearest 0.001.
- Using the example above, write an exponential expression and a radical expression to represent the amount of virus present for the following days:
a) \(\frac{1}{3}\) days
b) \(\frac{1}{4}\) days
c) \(\frac{5}{6}\) days
Check your solutions here.
