Respuesta :
Answer:
It will take 0.62 seconds for the paint ball to hit the disc.
Step-by-step explanation:
Height of the disk:
[tex]H_d = -5t^2 + 31.5t + 2[/tex]
Height of the paintball:
[tex]H_p = 30t + 1[/tex]
When the paintball will hit the disk?
When they are at the same height, so:
[tex]H_d = H_p[/tex]
[tex]-5t^2 + 31.5t + 2 = 30t + 1[/tex]
[tex]5t^2 - 1.5t - 1 = 0[/tex]
Solving a quadratic equation:
Given a second order polynomial expressed by the following equation:
[tex]ax^{2} + bx + c, a\neq0[/tex].
This polynomial has roots [tex]x_{1}, x_{2}[/tex] such that [tex]ax^{2} + bx + c = a(x - x_{1})*(x - x_{2})[/tex], given by the following formulas:
[tex]x_{1} = \frac{-b + \sqrt{\Delta}}{2*a}[/tex]
[tex]x_{2} = \frac{-b - \sqrt{\Delta}}{2*a}[/tex]
[tex]\Delta = b^{2} - 4ac[/tex]
In this question:
Quadratic equation with [tex]a = 5, b = -1.5, c = -1[/tex]
So
[tex]\Delta = (-1.5)^2 - 4(5)(-1) = 22.25[/tex]
[tex]t_{1} = \frac{-(-1.5) + \sqrt{22.25}}{2(5)} = 0.62[/tex]
[tex]t_{2} = \frac{-(-1.5) - \sqrt{22.25}}{2(5)} = -0.32[/tex]
Time is a positive measure, so 0.62.
It will take 0.62 seconds for the paint ball to hit the disc.
