A fountain on a lake sprays water in a parabolic arch modeled by the equation
y = −0.4x2 + 3x, where y is the height in feet and x is the horizontal distance. A beam of light is passed through the fountain to create a rainbow effect. If the beam is directed at an angle modeled by the equation -2.2x + 4.9y = 8.18, at what distance from the ground will the beam first touch the water spray?
y = −0.4x2 + 3x, where y is the height in feet and x is the horizontal distance. A beam of light is passed through the fountain to create a rainbow effect. If the beam is directed at an angle modeled by the equation -2.2x + 4.9y = 8.18, at what distance from the ground will the beam first touch the water spray?
You are required to find the y coordinate of the intersection point between the curve and the line assuming that the x axis is the ground
if you plot the two functions the line represented by the equation
-2.2x + 4.9y = 8.18 intersects the curve represented by the equation
y = −0.4x2 + 3x. To find the point of intersection:
rewrite equation -2.2x + 4.9y = 8.18 in terms of y
4.9y=8.18+2.2x, y= (8.18/4.9)+(2.2/4.9)x=1.67+0.45x
rearrange y=0.45x+1.67--eq1
y = -0.4x2+3x...eq2 Equate Eq1 to Eq2, y=y
0.45x+1.67=-0.4x2+3x, arrange terms
0.4x2+0.45x-3x+1.67=0, then
0.4x2-2.55x+1.67=0 This is a quadratic equation in x
solve a=0.4 b=-2.55 c=1.67
x=-b±√b2-4ac/2a = 2.55±√(-2.55)2-4(0.4)(1.67)/2(0.4)
x=2.55±√3.83/0.8, x=2.55±1.96/0.8 here you have two solutions
x=5.63, and x=0.73 Both Solutions are valid which means that the line intersects the fountain curve in two points but you are interested in the first point based on the problem statement that is x =0.73
From eq1 y=0.45x+1.67 solve for y = 0.45(0.73)+1.67=2.00 ft
which is the distance from the ground the beam first touch the water spray
Answer B