An arcade manager finds that revenue, R, based on a per-game fee, f, for unlimited play can be modeled by the function R=−480f^2+3,120f

Question

. Which of the following correctly interprets the average rate of change in revenue if the per-game fee increased from $1 to $3?(1 point)
Responses

Revenue would decrease by an average of $1,200 per $1 increase in the per-game fee.
Revenue would decrease by an average of $1,200 per $1 increase in the per-game fee.

The per-game fee would decrease by an average of $1,200 per $1 increase in the revenue.
The per-game fee would decrease by an average of $1,200 per $1 increase in the revenue.

Revenue would increase by an average of $1,200 per $1 increase in the per-game fee.
Revenue would increase by an average of $1,200 per $1 increase in the per-game fee.

The per-game fee would increase by an average of $1,200 per $1 increase in the revenue.

Answer 1

To determine the average rate of change in revenue when the per-game fee increases from $1 to $3, we first need to calculate the revenue at both fee levels using the given function \( R = -480f^2 + 3120f \).

1. Calculate Revenue at \( f = 1 \): \[ R(1) = -480(1^2) + 3120(1) = -480 + 3120 = 2640 \]

2. Calculate Revenue at \( f = 3 \): \[ R(3) = -480(3^2) + 3120(3) = -480(9) + 3120(3) = -4320 + 9360 = 5040 \]

3. Determine the Change in Revenue: \[ \text{Change in Revenue} = R(3) - R(1) = 5040 - 2640 = 2400 \]

4. Calculate the Change in Fee: \[ \text{Change in Fee} = 3 - 1 = 2 \]

5. Average Rate of Change: \[ \text{Average Rate of Change} = \frac{\text{Change in Revenue}}{\text{Change in Fee}} = \frac{2400}{2} = 1200 \]

Thus, when the per-game fee increases from $1 to $3, the average rate of change in revenue is $1,200 for every $1 increase in the per-game fee.

The correct interpretation is:

Revenue would increase by an average of $1,200 per $1 increase in the per-game fee.