Trapezoid Formed By Lines

The x-axis and the three lines x = 1, x = 4 and y = mx + 4 form a trapezoid. If the area of the trapezoid is 7, then what is m?

  1. -1/2
  2. -2/3
  3. -3/2
  4. -2
  5. none of these


Show/Hide Solution

Here’s the set-up.

A = (1, m ⋅ 1 + 4) = (1, m + 4), so b1 = m + 4
B = (4, 4m +4), so b2 = 4m + 4.

Area = ½(b1 + b2) ⋅ 3   = 3/2 (m + 4 + 4m + 4)
  = 3/2 (5m + 8) = 15m/2 + 12 = 7 (given).
  → 15m/2 = -5 → 15m = -10 → m = -2/3 (b).


So in the sketch, our line should have negative slope, but who knew? Well, OK, OK, we could have known. Suppose we had a rectangle, then m = 0, and b1 = m + 4 = 4. So the area of the rectangle would be 3 x 4 = 12, which is bigger than the area of the trapezoid. So m is negative already. Sheesh.

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