Question

Rob is a geologist. He is surveying a conical crater that was created by a meteor impact. From one end to another, the crater forms a V shape. The deepest part of the crater is 400 feet deep. The depth of the crater varies at a rate of 0.25 foot with the horizontal distance from the west end of the crater. The equation that models the depth of the crater, d, in feet, with respect to the horizontal distance in feet from the west end of the crater, h, is d = . The depth of the crater is 250 feet at horizontal distances of and feet from the west end.

Answer 1

Answer:

• equation: d(h) = 0.25|h -1600| -400
• depth is 250 ft at 1000 ft and 2200 ft from the west edge

Step-by-step explanation:

The crater shape can be modeled by an absolute value function with a slope of 0.25. The vertex of the function will not be at (0, 0) but will be at (1600, -400). The usual methods of translating functions apply. Horizontal displacement of the vertex is subtracted from the independent variable; vertical displacement is added to the function value.

  d(h) = 0.25×|h -1600| -400

We know the horizontal displacement is 1600 ft, because the depth changes at a rate of 1/4 foot for each horizontal foot. A depth change of 400 feet will require 1600 horizontal feet to accomplish.

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At a depth of 250 ft, the distance from the west edge can be found from ...

  -250 = 0.25|h -1600| -400

  150 = 0.25|h -1600| . . . . . . . . add 400

  600 = |h -1600| . . . . . . . . . . . multiply by 4

This resolves to two equations:

• -600 = h -1600   ⇒   h = 1000
• 600 = h -1600   ⇒   h = 2200

The depth is 250 ft at distances of 1000 ft and 2200 ft from the west edge.

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Comment on the equation

We have chosen to make depths be negative numbers. If you want the equation to give positive numbers for depth, multiply it by -1:

  d = 400 -0.25×|h -1600|