Question

1. the number of miles clare rode her bike this year is 50 miles. she plans to triple the number of miles she rides her bike each year.

a. use this information to complete the table.

number of years since this year	number of miles clare rides her bike
1
2
3
4

b. what do you notice about the difference in the number of miles clare is going to ride her bike from year to year?

c. what do you notice about the factor by which the number of miles clare rides her bike changes each year?

d. one year, clare rides ( m ) miles on her bike. how many miles will clare ride her bike the next year?

Explanation:

Part a

Step1: Year 0 (Current Year)

At year 0, the number of miles is the initial amount, which is 50 miles.

Step2: Year 1

To find the miles for year 1, we triple the miles from the previous year (year 0). So, \( 50\times3 = 150 \) miles.

Step3: Year 2

Triple the miles from year 1: \( 150\times3 = 450 \) miles.

Step4: Year 3

Triple the miles from year 2: \( 450\times3 = 1350 \) miles.

Step5: Year 4

Triple the miles from year 3: \( 1350\times3 = 4050 \) miles.

The completed table is:

number of years since this year	number of miles Clare rides her bike
1	150
2	450
3	1350
4	4050

Part b

We calculate the differences between consecutive years' miles: \( 150 - 50 = 100 \), \( 450 - 150 = 300 \), \( 1350 - 450 = 900 \), \( 4050 - 1350 = 2700 \). The differences are 100, 300, 900, 2700, which are increasing. So the difference in miles from year to year is not constant; it is increasing.

To find the factor, we divide the miles of a year by the miles of the previous year: \( \frac{150}{50} = 3 \), \( \frac{450}{150} = 3 \), \( \frac{1350}{450} = 3 \), \( \frac{4050}{1350} = 3 \). The factor by which the number of miles changes each year is constant (3).

Answer:

The difference in the number of miles from year to year is increasing (not constant).

Part c