Question

a golf tournament begins with 128 players. at the end of round one, there will be 64 players. there are 32 players at the end of round 2 and 16 players remaining at the end of round 3. which equation best models this situation?

\\(y = 128(2)^x\\)
\\(y = 0.5x + 128\\)
\\(y = -64x + 128\\)
\\(y = 128(0.5)^x\\)

Explanation:

Identify the initial value and key data points

The tournament begins with 128 players, so at round \(x = 0\), the number of players is \(y = 128\).
The remaining data points are:

• Round 1 (\(x = 1\)): \(y = 64\)
• Round 2 (\(x = 2\)): \(y = 32\)
• Round 3 (\(x = 3\)): \(y = 16\)

Determine the pattern of decay

Each round, the number of players is halved:

$$ \frac{64}{128} = 0.5,\quad \frac{32}{64} = 0.5,\quad \frac{16}{32} = 0.5 $$

This represents exponential decay with a base (decay factor) of \(0.5\).

Formulate the exponential equation

Using the standard exponential form \(y = a(b)^x\), where \(a\) is the initial value and \(b\) is the decay factor:

$$ y = 128(0.5)^x $$

Answer:

• (A) \(y = 128(2)^x\)
• (B) \(y = 0.5x + 128\)
• (C) \(y = -64x + 128\)
• (D) \(y = 128(0.5)^x\) (Correct answer)