Lesson 8
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.
Solution
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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.
Solution
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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 |
Solution
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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.
Solution
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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?
Solution
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Problem 6
How would you describe the output of this graph for:
- inputs from 0 to 1
- inputs from 3 to 4
Solution
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(From Unit 4, Lesson 1.)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 | |
\(2 \boldcdot 5730\) | |
\(3 \boldcdot 5730\) | |
\(4 \boldcdot 5730\) |
Solution
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(From Unit 4, Lesson 7.)