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Difficulty: MediumSlope of a Line

A water tank is being drained of its contents at a constant rate. The depth of the water in the tank is 8.28.2 feet after 22 hours of draining, and it is 5.85.8 feet after 66 hours of draining. What is the slope of the line that represents the depth of the water, in feet, as a function of the draining time, in hours?

  1. A
    −53-\frac{5}{3}
  2. −35-\frac{3}{5}Answer
  3. C
    −310-\frac{3}{10}
  4. D
    35\frac{3}{5}
  5. E
    53\frac{5}{3}

Answer

The slope of the line that represents the depth of the water as a function of the draining time is −35-\frac{3}{5}.
The correct answer is −35-\frac{3}{5}. To find the slope, we determine the points (t,d)(t, d) from the given information: (2,8.2)(2, 8.2) and (6,5.8)(6, 5.8). Using the slope formula m=d2−d1t2−t1m = \frac{d_2 - d_1}{t_2 - t_1}, we compute 5.8−8.26−2=−2.44=−0.6\frac{5.8 - 8.2}{6 - 2} = \frac{-2.4}{4} = -0.6. Converting −0.6-0.6 to a fraction gives −610-\frac{6}{10}, which simplifies to −35-\frac{3}{5}.

Step-by-Step Solution

1
Identify the coordinates from the problem statement.
The two coordinates (t,d)(t, d) where tt is time in hours and dd is depth in feet are (2,8.2)(2, 8.2) and (6,5.8)(6, 5.8).
Since the question asks for the depth as a function of time, time is the independent variable (xx) and depth is the dependent variable (yy).
2
Apply the slope formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.
m=5.8−8.26−2m = \frac{5.8 - 8.2}{6 - 2}
The slope represents the constant rate of change of the water level over time.
3
Simplify the fraction and convert to a simplified ratio.
m=−2.44=−0.6=−35m = \frac{-2.4}{4} = -0.6 = -\frac{3}{5}
Dividing the change in depth by the change in time yields the slope of the linear relationship.

Key Concept

Slope of a Line
Estimated Time:1m 0s
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Topics

Slope of a Line

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