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\dfrac{-1}{2}

  2. −23\dfrac{-2}{3}

  3. −32\dfrac{-3}{2}

  4. −2-2

  5. none of these

Solution

Take a look at the sketch below. For coordinates, we have A=(1,m+4)A = (1, m + 4) and B=(4,4m+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}\dfrac{1}{2} (b_1 + b_2) \cdot 3 &= \dfrac{3}{2} (m + 4 + 4m + 4) \\[1em]&= \dfrac{15m}{2} + 12 = 7\end{aligned}

Therefore, m=−23m = \dfrac{-2}{3}.