An archway is modeled by the equation y = -2x2 + 8x. A rod is to be placed across the archway at an angle defined by the equatio
n x − 2.23y + 10.34 = 0. If the rod is attached to the archway at points A and B, such that point B is at a higher level than point A, at what distance from the ground level is point B?
1) y= - 2x² + 8x. It's a parabola open downward (a<0) 2) x - 2.23.y + 10.34 = 0 . Re-write it : y = (x/2.23) + (10.34/2.23), a linear equation. To find the intersections between 1) & 2), let 1) = 2) -2x² + 8x = (x/2.23) + (10.34/2.23) -2x² + 8x - (x/2.23) - (10.34/2.23) =0 ; solve this quadratic for x values:
x' (that is A) = 0.772 & x" (that is B) = 3. (these are the values of x-intercept (parabola with line). To calculate the y-values, plug x' & x' in the equation: for x' = 0.772, y = 0.34 → B(0.772 , 0.34) for x" = 3, y = 0.016 → A(3 , 0.O16) So B IS AT 0.34 Unit from the ground
EXPLANATION To solve this problem you need to remember that a straight line is 180°. This figure is basically a straight line split in two, so 134°+x=180°. You can also subtract 134 from 180 to get 46.
A function is a relation for which each value from the set the first components of the ordered pairs is associated with exactly one value from the set of second components of the ordered pair.
