Question

A music competition on television had five elimination rounds. After each elimination, only half of the contestants were sent to the next round. The table below shows the number of contestants in each round of the competition:


x 1 2 3 4 5
f(x) 64 32 16 8 4


Compute the average rate of change of f(x) from x = 3 to x = 5, and describe what it represents.
a −6 contestants per round, and it represents the number of contestants who will reach the final round
b −28 contestants per round, and it represents the number of contestants who will reach the final round
c −6 contestants per round, and it represents the average rate at which the number of contestants decreased from the third round to the fifth round
d −28 contestants per round, and it represents the average rate at which the number of contestants decreased from the third round to the fifth round

Answer 1

To compute the average rate of change of \( f(x) \) from \( x = 3 \) to \( x = 5 \), we can use the formula for the average rate of change, which is given by:

\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]

where \( a = 3 \) and \( b = 5 \).

From the table:
- \( f(3) = 16 \)
- \( f(5) = 4 \)

Now, we can substitute these values into the formula:

\[
\text{Average Rate of Change} = \frac{f(5) - f(3)}{5 - 3} = \frac{4 - 16}{5 - 3} = \frac{-12}{2} = -6
\]

The average rate of change of \( f(x) \) from \( x = 3 \) to \( x = 5 \) is \( -6 \) contestants per round.

Now, let's interpret what this result represents. The negative value indicates that the number of contestants decreased as we move from the third round to the fifth round. Specifically, it means that, on average, 6 contestants were eliminated per round between the third and fifth rounds.

Given the options, the correct answer is:

**c −6 contestants per round, and it represents the average rate at which the number of contestants decreased from the third round to the fifth round.**