Let's solve each part step by step based on the information provided:

Given:

• Charlie inherited the collection in 1999 with 1,239 marbles.
• The collection started in 1905.
• In 1966, there were 810 marbles.
• Each year, the same number of marbles was added.

Part (a): For how many years had the collection been in existence by 1999?

The collection started in 1905, and Charlie inherited it in 1999. So, the number of years the collection was in existence by 1999 is: $1999 - 1905 = 94 \text{ years}$

Part (b): How many marbles are added to the collection each year?

Between 1966 (with 810 marbles) and 1999 (with 1,239 marbles), there is a difference of: $1239 - 810 = 429 \text{ marbles}$ The number of years between 1966 and 1999 is: $1999 - 1966 = 33 \text{ years}$ So, the number of marbles added each year is: $\frac{429}{33} \approx 13 \text{ marbles per year}$

Part (c): How many marbles were in the original collection in 1905?

From 1966 to 1905, the collection spans: $1966 - 1905 = 61 \text{ years}$ If 13 marbles were added each year, the number of marbles added from 1905 to 1966 is: $61 \times 13 = 793 \text{ marbles}$ So, the original number of marbles in 1905 was: $810 - 793 = 17 \text{ marbles}$

Part (d): Function describing the growth of the marble collection

Let $n$ be the number of years since 1905. The number of marbles, $M$, in the collection can be expressed as: $M(n) = 17 + 13n$ where $n = 0$ represents the year 1905.

Part (e): When will the collection have more than 2,000 marbles?

We need to solve for $n$ when $M(n) > 2000$: $17 + 13n > 2000$ $13n > 1983$ $n > \frac{1983}{13} \approx 152.54$ Rounding up, $n = 153$.

So, 153 years after 1905: $1905 + 153 = 2058$ The collection will have more than 2,000 marbles in the year 2058.

Would you like more details or have any questions?

Related Questions:

1. How would the function $M(n)$ change if the number of marbles added each year doubled?
2. What if the initial number of marbles in 1905 had been different? How would that affect the function?
3. How would you solve this problem if the number of marbles added each year changed over time?
4. What if Charlie inherited the collection at a different year, say 1980? How would the calculations change?
5. How can we model the situation if instead of adding marbles every year, marbles were added every five years?

Tip:

When solving problems involving linear growth, identifying the rate of change and initial value is crucial for formulating a linear equation.