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Zorluk: OrtaInterpreting Linear Relationships in Context

An electric delivery van's remaining battery capacity, BB, in kilowatt-hours (kWh), after driving dd miles is modeled by a linear equation. After driving 4545 miles, the remaining battery capacity is 6868 kWh. After driving a total of 120120 miles, the remaining battery capacity is 3838 kWh. According to this model, how many miles can the van drive on a full charge before the battery capacity reaches 00 kWh?

Cevap: 215 miles

Cevap

215
To find the maximum distance the van can drive on a full charge, we model the relationship between the remaining battery capacity, BB, and the distance driven, dd, using the linear equation B=md+B0B = md + B_0. First, calculate the rate of consumption (slope, mm) using the points (45,68)(45, 68) and (120,38)(120, 38): m=38−68120−45=−3075=−0.4m = \frac{38 - 68}{120 - 45} = \frac{-30}{75} = -0.4 kWh per mile. Next, find the initial battery capacity (B0B_0) by substituting the values from one of the points: 68=−0.4(45)+B068 = -0.4(45) + B_0, which simplifies to 68=−18+B068 = -18 + B_0, so B0=86B_0 = 86 kWh. Finally, determine the distance dd when the battery is fully depleted (B=0B = 0): 0=−0.4d+860 = -0.4d + 86, which yields 0.4d=860.4d = 86, and d=215d = 215 miles.

Adım Adım Çözüm

1
Calculate the rate of change of the battery capacity per mile driven.
The rate of change (slope) is −0.4-0.4 kWh per mile.
The slope mm represents the energy consumption rate and is calculated using the two data points (45,68)(45, 68) and (120,38)(120, 38) with the formula m=38−68120−45=−3075=−0.4m = \frac{38 - 68}{120 - 45} = \frac{-30}{75} = -0.4.
2
Determine the initial battery capacity when the distance driven is 0 miles.
The initial battery capacity is 8686 kWh.
Using the slope-intercept form B=md+B0B = md + B_0, substitute m=−0.4m = -0.4 and the point (45,68)(45, 68) to get 68=−0.4(45)+B068 = -0.4(45) + B_0. This simplifies to 68=−18+B068 = -18 + B_0, so B0=86B_0 = 86.
3
Solve for the distance driven dd when the battery capacity BB is 00 kWh.
The distance is 215215 miles.
Set B=0B = 0 in the linear model B=−0.4d+86B = -0.4d + 86 to get 0=−0.4d+860 = -0.4d + 86. Solving for dd gives 0.4d=860.4d = 86, which results in d=860.4=215d = \frac{86}{0.4} = 215.

Anahtar Kavram

Interpreting and calculating slope, y-intercept, and x-intercept of a linear function in a real-world context.
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Interpreting Linear Relationships in Context

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