Question

The path traveled by a bottlenose dolphin as it jumps out of water is modeled by the equation y = −0.4x2 + 3x, where y is the height above water and x is the horizontal distance in feet. If a beam of light is shone upward at an angle modeled by the equation x + y = 10, at what height from the water's surface will the beam of light hit the dolphin

Answer 1

The beam of light hit the dolphin at the common point that is the solution of a system:

`image`.

From the second equation express y: y=10-x and substitute it into the first equation:

`10-x=-0.4x^2+3x,\\ 10-x+0.4x^2-3x=0,\\ 0.4x^2-4x+10=0,\\ 4x^2-40x+100=0,\\ x^2-10x+25=0,\\ (x-5)^2=0,\\ x=5`.

Therefore, the horizontal distance in feet is x=5 ft and the height above water is y=10-5=5 ft.

Answer: the beam of light hit the dolphin at 5 ft above water