The problem describes the height of a rocket launched from a tower, given by the quadratic equation:

$y = -16t^2 + 112t + 17$

where $y$ is the height in feet and $t$ is the time in seconds after launch. The question asks for the time when the rocket will hit the ground, which corresponds to finding the time $t$ when $y = 0$.

Step 1: Set the equation equal to zero

To find the time when the rocket hits the ground, solve the equation:

$-16t^2 + 112t + 17 = 0$

Step 2: Use the quadratic formula

The quadratic formula is:

$t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In this equation, $a = -16$, $b = 112$, and $c = 17$. Plugging these values into the quadratic formula:

$t = \frac{-112 \pm \sqrt{112^2 - 4(-16)(17)}}{2(-16)}$

Step 3: Calculate the discriminant and solve

First, calculate the discriminant:

$\text{Discriminant} = 112^2 - 4(-16)(17) = 12544 + 1088 = 13632$

Now, find the square root of the discriminant:

$\sqrt{13632} \approx 116.71$

Substitute back into the quadratic formula:

$t = \frac{-112 \pm 116.71}{-32}$

This results in two possible solutions:

$t_1 = \frac{-112 + 116.71}{-32} \approx \frac{4.71}{-32} \approx -0.147 \text{ (not physically meaningful, as time cannot be negative)}$ $t_2 = \frac{-112 - 116.71}{-32} \approx \frac{-228.71}{-32} \approx 7.15$

Final Answer

The rocket will hit the ground approximately 7.15 seconds after launch.

Would you like a more detailed explanation or have any questions?

Here are 5 related questions to expand on this topic:

1. How does the initial height of the tower affect the time it takes for the rocket to hit the ground?
2. What is the maximum height reached by the rocket, and at what time does it occur?
3. How would the time change if the initial velocity of the rocket were doubled?
4. Can you explain the significance of the discriminant in the quadratic equation?
5. What would happen if the acceleration due to gravity were different?

Tip: Always check the discriminant before solving a quadratic equation to determine the nature of the roots (real and distinct, real and repeated, or complex).