2761.13 – A Tantalizing Trapezoid

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

  1. −12\frac{-1}{2}
  2. −23\frac{-2}{3}
  3. −32\frac{-3}{2}
  4. −2-2
  5. none of these

2761_13_4ba8a92ae4.png

Solution

Take a look at the sketch below. For coordinates, we have:

A=(1,m⋅1+4)=(1,m+4),B=(4,4m+4).A = (1, m\cdot1 + 4) = (1, m + 4), B = (4, 4m + 4).

So, b1=m+4b_1 = m+4 and b2=4m+4.b_2 = 4m + 4. For the area we have,

12(b1+b2)⋅3=32(m+4+4m+4)=15m2+12=7.\begin{aligned} \frac{1}{2} (b_1 + b_2) \cdot 3 &=& \frac{3}{2} (m + 4 + 4m + 4) \\ &=& \frac{15m}{2} + 12 = 7. \end{aligned}

Therefore, m=−23m = \frac{-2}{3}.

2761_13_solution_8052504e3a.png