Due In 6 Hours: Short Answer, No Essay Required Rational Fun

Due In 6 Hours Short Answer No Essay Requiredrational Function

Find the cost of a device, research a monthly plan, and model the monthly cost as a rational function. Analyze the asymptotes for minimum average monthly cost, and calculate the average cost after 24 months. Explain how this influences your perception of phone pricing.

Paper For Above instruction

The modeling of the monthly cost of a smartphone purchased upfront, combined with a monthly plan, can be effectively represented through a rational function. This approach helps to understand how the total costs distribute over time, and specifically, reveals the behavior of average costs as the ownership period extends. To explore this, I selected a recent smartphone model priced at $800, and a typical monthly plan costing $50 for data, calls, and messages. These concrete numbers provide an empirical basis for constructing our model.

Assuming the total cost comprises the device's upfront cost spread over time (m months) and the monthly plan fees, the total cost after m months is given by:

Total cost (C(m)) = \(\displaystyle 800 + 50m\)

To model the average monthly cost, which is what consumers often consider, we divide the total cumulative expenses by the number of months m. This yields the rational function:

Average monthly cost (A(m)) = \(\displaystyle \frac{C(m)}{m} = \frac{800 + 50m}{m} = \frac{800}{m} + 50\)

This function reflects how the initial device cost spreads out over time, diminishing as m increases, while the monthly plan cost remains constant. The asymptote as m approaches infinity is y = 50, indicating that with very long ownership durations, the average monthly cost approaches the monthly plan fee alone. This is intuitive because the initial investment becomes negligible over extensive ownership periods.

The asymptote at m → 0 is a vertical asymptote, which signifies that in the first month, the average monthly cost is very high due to the entire device cost being attributed to a single month. The minimum average monthly cost occurs as m approaches infinity, tending toward $50, the ongoing monthly plan fee.

To determine the minimum average monthly cost explicitly, we observe that as ownership duration increases, the average cost decreases asymptotically towards $50. In practical terms, the lowest average cost is achieved after owning the phone for a sufficiently long period, often beyond the typical two-year plan, thus making the cost seem more affordable over time.

Calculating the average cost after 24 months (2 years), we substitute m = 24 into our function:

\(\displaystyle A(24) = \frac{800}{24} + 50 \approx 33.33 + 50 = 83.33\)

This means that after two years, the average monthly expense associated with this phone and plan is approximately $83.33. Comparing this to the initial costs, consumers might reconsider whether paying upfront or in installments affects their overall affordability.

From this analysis, it is evident that the longer the ownership period, the more the initial device cost influences the average monthly expense. The decreasing trend towards the asymptote suggests that spreading out the device cost over a longer period reduces the average monthly cost, which could influence how consumers perceive the affordability of expensive phones. While a short-term perspective might see the high initial cost as prohibitive, understanding that costs diminish over time encourages a long-term view—highlighting the importance of considering total costs spread out over the ownership period.

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