Question

Earlier, you were told about a toy rocket fired into the air from the top of a barn. Its height (h) above the
ground in yards after a seconds is given by the function:
h(x) = -52% + 100 + 20
What was the maximum height of the rocket?
Solution
The maximum height was reached by the rocket at one second as you found in part b from the previous
example. It takes second to reach the maximum height. We will substitute that value in for x in our
function and simplify.
The maximum height reached by the rocket was 25 yards. What is the time it takes for the rocket to hit
the ground? (Use a graph or any other method to solve.)
It takes approximately 3.2 seconds for the rocket to hit the ground. (Round to the nearest tenth.)

Answer 1

Using quadratic function concepts, it is found that:

• It takes 1 second to reach maximum height.
• The maximum height is of 25 yards.

• It takes approximately 3.2 seconds for the rocket to hit the ground.

The height of the rocket after x seconds is given by:

$h(x) = -5x^2 + 10x + 20$

Which is a quadratic function with coefficients $a = -5, b = 10, c = 20$.

The maximum height will be a reached at a time of $x_v$, and will be of $y_v$, considering the vertex given by:

$x_v = -\frac{b}{2a}$

$y_v = -\frac{\Delta}{4a}$

$\Delta = b^2 - 4ac$

Then:

$x_v = -\frac{10}{2(-5)} = 1$

$\Delta = 10^2 - 4(-5)(20) = 500$

$y_v = -\frac{500}{4(-5)} = 25$

Thus:

• It takes 1 second to reach maximum height.
• The maximum height is of 25 yards.

It hits the ground when $h(x) = 0$, thus:

$-5x^2 + 10x + 20 = 0$

$x_1 = \frac{-b + \sqrt{\Delta}}{2a} = \frac{-10 + \sqrt{500}}{2(-5)} = -1.2$

$x_2 = \frac{-b - \sqrt{\Delta}}{2a} = \frac{-10 - \sqrt{500}}{2(-5)} = -3.2$

We want the positive root, so, it takes approximately 3.2 seconds for the rocket to hit the ground.

A similar problem is given at https://brainly.com/question/24705734