Lesson 8
Unknown Exponents
- Let’s find unknown exponents.
Problem 1
A pattern of dots grows exponentially. The table shows the number of dots at each step of the pattern.
| step number | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| number of dots | 1 | 5 | 25 | 125 |
- Write an equation to represent the relationship between the step number, \(n\), and the number of dots, \(y\).
- At one step, there are 9,765,625 dots in the pattern. At what step number will that happen? Explain how you know.
Problem 2
A bacteria population is modeled by the equation \(p(h) = 10,\!000 \boldcdot 2^h\), where \(h\) is the number of hours since the population was measured.
About how long will it take for the population to reach 100,000? Explain your reasoning.
Problem 3
Complete the table.
| \(x\) | -2 | 0 | \(\frac{1}{3}\) | 1 | ||||
|---|---|---|---|---|---|---|---|---|
| \(10^x\) | \(\frac{1}{10,000}\) | \(\frac{1}{1,000}\) | \(\frac{1}{100}\) | \(\hspace{.6cm}\) | \(\hspace{.6cm}\) | \(\hspace{.6cm}\) | 1,000 | 1,000,000,000 |
Problem 4
Here is a graph of \(y = 3^x\).
What is the approximate value of \(x\) satisfying \(3^x = 10,\!000\)? Explain how you know.
Problem 5
One account doubles every 2 years. A second account triples every 3 years. Assuming the accounts start with the same amount of money, which account is growing more rapidly?
Problem 6
How would you describe the output of this graph for:
- inputs from 0 to 1
- inputs from 3 to 4
Problem 7
The half-life of carbon-14 is about 5730 years.
- Complete the table, which shows the amount of carbon-14 remaining in a plant fossil at the different times since the plant died.
- About how many years will it be until there is 0.1 picogram of carbon-14 remaining in the fossil? Explain how you know.
| years | picograms |
|---|---|
| 0 | 3 |
| 5730 | |
| \(2 \boldcdot 5730\) | |
| \(3 \boldcdot 5730\) | |
| \(4 \boldcdot 5730\) |