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Difficulty: MediumSlope of a Line

A commercial drone starts at an altitude of 150150 meters and descends at a constant rate. After 1212 seconds of descent, its altitude is 114114 meters. If the altitude, aa, in meters, is modeled as a linear function of time, tt, in seconds, what is the slope of the line representing this relationship in the standard (t,a)(t, a) coordinate plane?

  1. A
    −36-36
  2. B
    −13-\frac{1}{3}
  3. −3-3Answer
  4. D
    13\frac{1}{3}
  5. E
    33

Answer

The slope of the line representing this relationship is −3-3.
The slope of the line represents the rate of change of the altitude with respect to time. By taking the initial point (0,150)(0, 150) and the point after 1212 seconds (12,114)(12, 114), the change in altitude is 114−150=−36114 - 150 = -36 meters. Dividing this by the change in time of 12−0=1212 - 0 = 12 seconds yields the correct slope of −3-3.

Step-by-Step Solution

1
Identify the coordinate points (t,a)(t, a) from the given information.
The initial state corresponds to the point (0,150)(0, 150), and the state after 1212 seconds corresponds to (12,114)(12, 114).
Setting up coordinate pairs allows the direct application of the slope formula.
2
Apply the slope formula m=a2−a1t2−t1m = \frac{a_2 - a_1}{t_2 - t_1} to find the rate of change.
m=114−15012−0m = \frac{114 - 150}{12 - 0}
The slope is the change in the vertical variable (altitude) divided by the change in the horizontal variable (time).
3
Calculate the difference and simplify the fraction.
m=−3612=−3m = \frac{-36}{12} = -3
Subtracting 150150 from 114114 gives the net change in altitude, which when divided by the duration of 1212 seconds yields the rate of change per second.

Key Concept

The slope of a line represents its constant rate of change, defined as the ratio of the change in the dependent variable to the change in the independent variable.
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Topics

Slope of a Line

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