how do I solve this word problem?

Question

A firefighter is on the fifth floor of an office building. She needs to throw a rope into the window above her on the seventh floor. The function h = -16r² + 36t  models how high above her she is able to throw a rope. If she needs to throw the rope 40ft above her to reach the seventh floor window, will the rope get to the window?

Answer 1

Set h(t) = 40 ft and solve for t.  If the solution for t is real, the rope can reach the window.  If the solution is complex, the rope cannot reach the window.

 

40 = -16t² + 36t

 

-16t² + 36t - 40 = 0 (divide through by 4 to simplify this a little)

 

-4t² + 9t - 10 = 0

 

Evaluate the discriminant of this equation.

 

D = b² - 4ac = 9² - 4(-4)(-10) = -79 < 0 

 

D < 0 means the solutions for t are complex numbers, and therefore the rope will not reach the window.