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

It takes 4 seconds for the projectile to hit the ground

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\\eq0`.

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) = (-b + √(\bigtriangleup))/(2*a)`

`x_(2) = (-b - √(\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) = (-32 + √(9216))/(2*(-16)) = -2`

`t_(2) = (-32 - √(9216))/(2*(-16)) = 4`

Time is a positive measure, so:

It takes 4 seconds for the projectile to hit the ground