Question

The height of a projectile launched upward at a speed of 32 feet/second from a height of 128 feet is given by the function h(t) = -16t^2 + 32t +128. How long will it take the projectile to hit the ground?

Answer 1

Answer:

It takes 4 seconds for the projectile to hit the ground

Step-by-step explanation:

The height of the projectile after t seconds is given by the following equation:

$h(t) = -16t^{2} + 32t + 128$

How long will it take the projectile to hit the ground?

It happens when $h(t) = 0$

So

$h(t) = -16t^{2} + 32t + 128$

$-16t^{2} + 32t + 128 = 0$

Solving a quadratic equation:

Given a second order polynomial expressed by the following equation:

$ax^{2} + bx + c, a\neq0$.

This polynomial has roots $x_{1}, x_{2}$ such that $ax^{2} + bx + c = a(x - x_{1})*(x - x_{2})$, given by the following formulas:

$x_{1} = \frac{-b + \sqrt{\bigtriangleup}}{2*a}$

$x_{2} = \frac{-b - \sqrt{\bigtriangleup}}{2*a}$

$\bigtriangleup = b^{2} - 4ac$

In this question:

$-16t^{2} + 32t + 128 = 0$

So $a = -16, b = 32, c = 128$

$\bigtriangleup = 32^{2} - 4*(-16)*(128) = 9216$

$t_{1} = \frac{-32 + \sqrt{9216}}{2*(-16)} = -2$

$t_{2} = \frac{-32 - \sqrt{9216}}{2*(-16)} = 4$

Time is a positive measure, so:

It takes 4 seconds for the projectile to hit the ground