An arcade manager finds that revenue, R, based on a per-game fee, f, for unlimited play can be modeled by the function R=−480f2+3,120f. 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)ResponsesRevenue 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.The per-game fee would increase by an average of $1,200 per $1 increase in the revenue.The per-game fee would increase 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.

Question

An arcade manager finds that revenue, R, based on a per-game fee, f, for unlimited play can be modeled by the function R=−480f2+3,120f. 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)ResponsesRevenue 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.The per-game fee would increase by an average of $1,200 per $1 increase in the revenue.The per-game fee would increase 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.

GPT-4o mini · Answer

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 points using the given revenue function:

\[ R(f) = -480f^2 + 3120f \]

Calculate $ R(1) $ (revenue when fee is $1): \[ R(1) = -480(1^2) + 3120(1) = -480 + 3120 = 2640 \]
Calculate $ R(3) $ (revenue when fee is $3): \[ R(3) = -480(3^2) + 3120(3) = -480(9) + 3120(3) = -4320 + 9360 = 5040 \]

Now, we can find the average rate of change in revenue between $ f = 1 $ and $ f = 3 $: \[ \text{Average Rate of Change} = \frac{R(3) - R(1)}{3 - 1} = \frac{5040 - 2640}{2} = \frac{2400}{2} = 1200 \]

This indicates that revenue increases by an average of $1,200 for each $1 increase in the per-game fee.

Therefore, the correct interpretation is:

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