# 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.

A:

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

B:

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

C:

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

D:

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

### 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.

### 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:

1. $$t$$, time in years
2. $$d$$, time in days

### 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

(From Unit 4, Lesson 5.)

### 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.

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

### Solution

(From Unit 4, Lesson 6.)

### Problem 6

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

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

### Solution

(From Unit 4, Lesson 6.)

### Problem 7

Select all expressions that are equal to $$8^{\frac23}$$.

A:

$$\sqrt[3]{8^2}$$

B:

$$\sqrt[3]{8}^2$$

C:

$$\sqrt{8^3}$$

D:

$$2^2$$

E:

$$2^3$$

F:

4