Question

3) Cory is standing on a cliff that is 48 feet above the ground. He throws a rock in to the air. The height, h, of the rock

Answer 1

Answer:

a) The initial height of the rock is of 48 feet.

b) It takes 1 seconds for the rock to reach maximum height.

c) The rock's maximum height is 50 feet.

d) It takes 6 seconds for the rock to land on the ground.

Step-by-step explanation:

Vertex of a quadratic function:

Suppose we have a quadratic function in the following format:

$f(x) = ax^{2} + bx + c$

It's vertex is the point $(x_{v}, f(x_{v})$

In which

$x_{v} = -\frac{b}{2a}$

If a<0, the vertex is a maximum point, that is, the maximum value happens at $x_{v}$, and it's value is $f(x_{v})$

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:

The height of the ball after t seconds, in feet, is given by:

$h(t) = -2t^2 + 4t + 48$

Which is a quadratic equation with $a = -2, b = 4, c = 48$

a) What is the initial height?

This is $h(0) = 48$. So the initial height of the rock is of 48 feet.

b) How many seconds does it take to reach the maximum height?

This is the value of t at the vertex. So

$t_{v} = -\frac{b}{2a} = -\frac{4}{2(-2)} = 1$

It takes 1 seconds for the rock to reach maximum height.

c) What is the rock's maximum height?

This is the height of the ball after 1 second. So

$h(1) = -2(1)^2 + 4(1) + 48 = 50$

The rock's maximum height is 50 feet.

d) How many seconds does it take to for the rock to land on the ground?

This is t for which $h(t) = 0$, so we solve the quadratic equation.

$\bigtriangleup = (4)^2-4(-2)(48) = 400$

$t_{1} = \frac{-4 + \sqrt{400}}{2*(-2)} = -4$

$t_{2} = \frac{-4 - \sqrt{400}}{2*(-2)} = 6$

Time is a positive measures, so it takes 6 seconds for the rock to land on the ground.