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Difficulty: Very hardInterpreting Linear Relationships in Context

For a wireless sensor node in an environmental monitoring network, the battery life BB, in days, when transmitting nn data packets per day is modeled by a linear function. Under standard operating conditions, the node's battery lasts for 120120 days when transmitting 8080 packets per day, and it lasts for 9090 days when transmitting 140140 packets per day. Under a new energy-saving firmware, the battery consumption rate per packet transmitted is reduced by 40%40\%, and the static power draw is also reduced, increasing the maximum battery life (when zero packets are transmitted) by 10%10\%. Under this new firmware, what is the battery life, in days, of a sensor node that transmits 150150 data packets per day?

Answer: 131 days

Answer

131
To solve this problem, we first determine the linear relationship B=mn+B0B = mn + B_0 under standard operating conditions. The rate of change (slope mm) represents the change in battery life per packet transmitted per day: m=90−120140−80=−0.5m = \frac{90 - 120}{140 - 80} = -0.5 days per packet. Substituting this back into the linear equation gives 120=−0.5(80)+B0120 = -0.5(80) + B_0, which simplifies to 120=−40+B0120 = -40 + B_0, yielding a y-intercept of B0=160B_0 = 160 days. Under the energy-saving firmware, the battery consumption rate per packet is reduced by 40%40\%. This means the rate at which battery life decreases per packet changes from 0.50.5 to 0.5×(1−0.40)=0.30.5 \times (1 - 0.40) = 0.3, giving us a new slope of −0.3-0.3. The maximum battery life (y-intercept) increases by 10%10\%, making the new y-intercept 160×1.10=176160 \times 1.10 = 176. The new linear model is Bnew=−0.3n+176B_{\text{new}} = -0.3n + 176. Evaluating this equation for n=150n = 150 packets per day yields Bnew=−0.3(150)+176=−45+176=131B_{\text{new}} = -0.3(150) + 176 = -45 + 176 = 131 days.

Step-by-Step Solution

1
Determine the linear relationship representing standard operating conditions by using the two given points, (80,120)(80, 120) and (140,90)(140, 90).
Slope m=−0.5m = -0.5 days per packet and y-intercept B0=160B_0 = 160 days.
Establishing the initial linear equation is necessary to obtain the base battery consumption rate and maximum battery capacity.
2
Calculate the new slope and y-intercept parameters under the energy-saving firmware by applying the specified percentage changes.
New slope mnew=−0.3m_{\text{new}} = -0.3 days per packet and new y-intercept B0,new=176B_{0,\text{new}} = 176 days.
To model the linear relationship under the updated power management firmware.
3
Formulate the new linear function Bnew=−0.3n+176B_{\text{new}} = -0.3n + 176 and substitute 150150 for nn.
Battery life B=131B = 131 days.
To find the expected battery life at the target daily packet transmission rate.

Key Concept

Interpreting how physical rates and initial values map to the slope and y-intercept of a linear model, and applying transformations to these parameters.
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Interpreting Linear Relationships in Context

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