# Lesson 7

Interpreting and Using Exponential Functions

### Problem 1

The half-life of carbon-14 is about 5,730 years. A fossil had 6 picograms of carbon-14 at one point in time. (A picogram is a trillionth of a gram or \(1 \times 10^{\text-12}\) gram.) Which expression describes the amount of carbon-14, in picograms, \(t\) years after it was measured to be 6 picograms.

\(6 \boldcdot \left(\frac12\right)^{\frac{t}{5,730}}\)

\(6 \boldcdot \left(\frac12\right)^{5,730t}\)

\(6 \boldcdot (5,\!730)^{\frac12 t}\)

\(\frac12 \boldcdot (6)^{\frac{t}{5,730}}\)

### Solution

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### Problem 2

The half-life of carbon-14 is about 5,730 years. A tree fossil was estimated to have about 4.2 picograms of carbon-14 when it died. (A picogram is a trillionth of a gram.) The fossil now has about 0.5 picogram of carbon-14. About how many years ago did the tree die? Show your reasoning.

### Solution

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### Problem 3

Nickel-63 is a radioactive substance with a half-life of about 100 years. An artifact had 9.8 milligrams of nickel-63 when it was first measured. Write an equation to represent the mass of nickel-63, in milligrams, as a function of:

- \(t\), time in years
- \(d\), time in days

### Solution

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### Problem 4

Tyler says that the function \(f(x) = 5^x\) is exponential and so it grows by equal factors over equal intervals. He says that factor must be \(\sqrt[10]{5}\) for an interval of \(\frac{1}{10}\) because ten of those intervals makes an interval of length 1. Do you agree with Tyler? Explain your reasoning.

### Solution

### Problem 5

The population in a city is modeled by the equation \(p(d) = 100,\!000 \boldcdot (1+ 0.3)^d\), where \(d\) is the number of decades since 1970.

- What do the 0.3 and 100,000 mean in this situation?
- Write an equation for the function \(f\) to represent the population \(y\) years after 1970. Show your reasoning.
- Write an equation for the function \(g\) to represent the population \(c\) centuries after 1970. Show your reasoning.

### Solution

### Problem 6

The function \(f\) is exponential. Its graph contains the points \((0,5)\) and \((1.5,10)\).

- Find \(f(3)\). Explain your reasoning.
- Use the value of \(f(3)\) to find \(f(1)\). Explain your reasoning.
- What is an equation that defines \(f\)?

### Solution

### Problem 7

Select **all **expressions that are equal to \(8^{\frac23}\).

\(\sqrt[3]{8^2}\)

\(\sqrt[3]{8}^2\)

\(\sqrt{8^3}\)

\(2^2\)

\(2^3\)

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