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3-42.

In 1999, Charlie received the family heirloom marble collection consisting of $1239$ marbles. Charlie’s great‑grandfather had started the original marble collection in 1905. Each year, Charlie’s great‑grandfather had added the same number of marbles to his collection. When he passed them on to his son, he insisted that each future generation add the same number of marbles per year to the collection. When Charlie’s father received the collection in 1966, there were $810$ marbles.

1. By the time Charlie inherited the collection, for how many years had it been in existence?

The collection had been in existence for $94$ years.

2. How many marbles are added to the collection each year?

The situation gives two data points from which to extrapolate this information: there were $810$ marbles in 1966, and $1239$ marbles in 1999.

$13$ marbles are added each year.

3. Use the information you found in part (b) to figure out how many marbles were in the original collection when Charlie’s great‑grandfather started it.

How many years passed between the start of the collection and when Charlie or his father received the collection?

How many marbles would have been added over all of those years?

4. Generalize this situation by writing a function describing the growth of the marble collection for each year (n) since Charlie’s great‑grandfather started it.

What kind of relation is this situation?

Use the information from parts (b) and (c) to construct the function.

5. How old will the marble collection be when Charlie (or one of his children) has more than 2000 marbles? In what year will this occur?

• Take the function you wrote in part (d) and set the number of marbles to $2000$, then solve for the number of years.

Enter the given data into the table in the eTool below.
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