Question

A rock is thrown upward from the top of a high cliff overlooking the ocean at a speed of 96 feet per second . The rock's height above ocean can be modeled by the equation H(t) = - 16t ^ 2 + 96t + 112 . How long will it take seconds for the rock to hit the ocean? Do not include units in your answer.

Answer 1

7

Explanation:

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 + √(\Delta))/(2*a)`

`x_(2) = (-b - √(\Delta))/(2*a)`

`\Delta = b^(2) - 4ac`

In this question:

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

`H(t) = -16t^2 + 96t + 112`

How long will it take in seconds for the rock to hit the ocean?

This is t for which:

`H(t) = 0`

So

`-16t^2 + 96t + 112 = 0`

Simplifying by -16

`t^2 - 6t - 7 = 0`

So a quadratic equation with
`a = 1, b = -6, c = -7`

`\Delta = b^(2) - 4ac = (-6)^2 - 4*1*(-7) = 36 + 28 = 64`

`t_(1) = (-(-6) + √(64))/(2*1) = 7`

`t_(2) = (-(-6) - √(64))/(2*1) = -1`

Since time is a positive measure, it will take 6 seconds for the rock to hit the ocean, and the answer is 7.