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A straight line of the form y = mx divides into two equal parts the region bounded by the y-axis, the horizontal line y = 3/2, and the parabola y = 1 + 4x - x². What is the value of m?
- 13/6
- 13/2
- 13/5
- 13/7
Correct answer: 13/6
Solution
The region is bounded by x=0, y=0, x=3/2 and the parabola y=1+4x-x^2, whose total area is the integral from 0 to 3/2 of (1+4x-x^2)=39/8. A line y=mx (which lies below the parabola) cuts off area m*(3/2)^2/2=9m/8; setting this equal to half, 9m/8=39/16, gives m=13/6.
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