describes how many people are remaining in the tournament after $$ x $$ rounds. How many people are left in the toumament after 2 rounds? $$ x=2 $$ $$ y=16\left(\frac{1}{2}\right)^{x} $$ INVESTMENTS Determine the amount of an investment if $$ \$ 1000 $$ is invested at an interest rate of $$ 8 \% $$ compo quarterly for 2 years.

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UpStudy AI · Accepted Answer

We are given two separate problems:

1. Tournament Remaining Players:
   The formula is y = 16 · (1/2)^x. For x = 2 rounds:
   • y = 16 · (1/2)^2 = 16 · (1/4) = 4.
   So, there are 4 people left in the tournament after 2 rounds.

2. Investment Calculation:
   The investment formula for compound interest is:
   A = P · (1 + r/n)^(n·t)
   where:
   • P = $1000 (initial investment)
   • r = 8% per year = 0.08
   • n = 4 (compounded quarterly)
   • t = 2 years

Substitute the values:
   A = 1000 · (1 + 0.08/4)^(4·2)
     = 1000 · (1 + 0.02)^8
     = 1000 · (1.02)^8

Now, calculate (1.02)^8. Using a calculator, we find:
   (1.02)^8 ≈ 1.17166

Therefore:
   A ≈ 1000 · 1.17166 ≈ $1171.66

Summary:
• After 2 rounds in the tournament, 4 players remain.
• The $1000 investment grows to approximately $1171.66 after 2 years.
After 2 rounds in the tournament, there are 4 players remaining. The $1000 investment grows to approximately $1171.66 after 2 years.

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