# Lesson 4

Understanding Decay

Let’s look at exponential decay.

### Problem 1

A new bicycle sells for $300. It is on sale for \(\frac{1}{4}\) off the regular price. Select **all** the expressions that represent the sale price of the bicycle in dollars.

\(300 \boldcdot \frac{1}{4}\)

\(300 \boldcdot \frac{3}{4}\)

\(300 \boldcdot \left(1 - \frac{1}{4}\right)\)

\(300 - \frac{1}{4}\)

\(300 - \frac{1}{4} \boldcdot 300\)

### Problem 2

A computer costs $800. It loses \(\frac{1}{4}\) of its value every year after it is purchased.

- Complete the table to show the value of the computer at the listed times.
- Write an equation representing the value, \(v\), of the computer, \(t\) years after it is purchased.
- Use your equation to find \(v\) when \(t\) is 5. What does this value of \(v\) mean?

time (years) |
value of computer (dollars) |
---|---|

0 | |

1 | |

2 | |

3 | |

\(t\) |

### Problem 3

A piece of paper is folded into thirds multiple times. The area, \(A\), of the piece of paper in square inches, after \(n\) folds, is \(A = 90 \boldcdot \left(\frac{1}{3}\right)^n\).

- What is the value of \(A\) when \(n = 0\)? What does this mean in the situation?
- How many folds are needed before the area is less than 1 square inch?
- The area of another piece of paper in square inches, after \(n\) folds, is given by \(B = 100 \boldcdot \left(\frac{1}{2}\right)^n\). What do the numbers 100 and \(\frac{1}{2}\) mean in this situation?

### Problem 4

At the beginning of April, a colony of ants has a population of 5,000.

- The colony decreases by \(\frac{1}{5}\) during April. Write an expression for the ant population at the end of April.
- During May, the colony decreases again by \(\frac{1}{5}\) of its size. Write an expression for the ant population at the end of May.
- The colony continues to decrease by \(\frac{1}{5}\) of its size each month. Write an expression for the ant population after 6 months.

### Problem 5

Lin has 13 mystery novels. Each month, she gets 2 more. Select **all** expressions that represent the total number of Lin's mystery novels after 3 months.

13 + 2 + 2+ 2

\(13 \boldcdot 2 \boldcdot 2 \boldcdot 2\)

\(13 \boldcdot 8\)

13 + 6

19

### Problem 6

An *odometer* is the part of a car's dashboard that shows the number of miles a car has traveled in its lifetime. Before a road trip, a car odometer reads 15,000 miles. During the trip, the car travels 65 miles per hour.

- Complete the table.
- What do you notice about the differences of the odometer readings each hour?
- If the odometer reads \(n\) miles at a particular hour, what will it read one hour later?

duration of trip (hours) |
odometer reading (miles) |
---|---|

0 | |

1 | |

2 | |

3 | |

4 | |

5 |

### Problem 7

A group of students is collecting 16 oz and 28 oz jars of peanut butter to donate to a food bank. At the end of the collection period, they donated 1,876 oz of peanut butter and a total of 82 jars of peanut butter to the food bank.

- Write a system of equations that represents the constraints in this situation. Be sure to specify the variables that you use.
- How many 16 oz jars and how many 28 oz jars of peanut butter were donated to the food bank? Explain or show how you know.

### Problem 8

A function multiplies its input by \(\frac 3 4\) then adds 7 to get its output. Use function notation to represent this function.

### Problem 9

A function is defined by the equation \(f(x) = 2x - 5\).

- What is \(f(0)\)?
- What is \(f(\frac 1 2)\)?
- What is \(f(100)\)?
- What is \(x\) when \(f(x) = 9\)?