Question

You find a brass bottle that looks really old. When you rub some dirt off of the bottle, a genie appears! The genie offers you a reward. You must choose one:$50,000, or A magical $1 coin. The coin will turn into two coins on the first day. The two coins will turn into four coins on the second day. The four coins will double to 8 coins on the third day. The genie explains the doubling will continue for 28 days.

1.How many days would it take for the number of magical coins to exceed $50,000?
2.Will the value of the magical coins exceed a million dollars within the 28 days? Explain or show your reasoning.

Answer 1

It takes just over 16 days for the magical coins to exceed $50,000, and by day 28, the value of the coins will exceed $268 million, which is far greater than $1 million.

Step-by-step explanation:

To determine how many days it would take for the number of magical coins to exceed $50,000, we use the formula for exponential growth:

Total = Initial amount * (2^(number of days)).

In this case, the initial amount is 1 (the magical $1 coin), and the total amount we want to exceed is $50,000.

1 * (2^days) > 50,000.

To find when this occurs, we need to solve for the days:

2^days > 50,000.

Days > log2(50,000).

Days > 15.6096.

It takes just over 16 days for the magical coins to exceed $50,000 in value.

To answer the second question, on day 28, we calculate the total value:

Total value = 1 * (2^28) = 268,435,456.

Yes, the value of the magical coins will definitely exceed a million dollars within 28 days, as it is significantly more than $1,000,000.