To find the time when the rocket will hit the ground, we need to determine when the height $y$ equals zero. This corresponds to the point when the rocket reaches the ground.

The equation given for the height $y$ is:

$y = -16x^2 + 231x + 71$

We set $y = 0$ to find the time $x$ when the rocket hits the ground:

$0 = -16x^2 + 231x + 71$

This is a quadratic equation in the form $ax^2 + bx + c = 0$, where:

• $a = -16$
• $b = 231$
• $c = 71$

We will solve for $x$ using the quadratic formula, which is:

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

Step 1: Calculate the discriminant

The discriminant is the part under the square root, $b^2 - 4ac$:

$b^2 - 4ac = 231^2 - 4(-16)(71)$

$b^2 - 4ac = 53361 + 4544$

$b^2 - 4ac = 57905$

Step 2: Apply the quadratic formula

Now, we can compute the values for $x$:

$x = \frac{-231 \pm \sqrt{57905}}{2(-16)}$

The square root of $57905$ is approximately $240.58$. So,

$x = \frac{-231 \pm 240.58}{-32}$

Step 3: Calculate the two possible values for $x$

1. Using the positive root:

$x_1 = \frac{-231 + 240.58}{-32} = \frac{9.58}{-32} \approx -0.30$

2. Using the negative root:

$x_2 = \frac{-231 - 240.58}{-32} = \frac{-471.58}{-32} \approx 14.73$

Since the negative time doesn't make sense in this context, we discard $x_1$ and keep the positive value, which is:

$x \approx 14.73 \, \text{seconds}$

Final Answer:

The rocket will hit the ground after approximately 14.73 seconds.

Would you like more details on the solution, or have any questions about this process?

Related Questions:

1. How would the solution change if the rocket's initial height were different?
2. What would happen if the rocket's initial velocity were lower or higher?
3. How do you solve quadratic equations when the discriminant is negative?
4. How can we find the maximum height of the rocket from the equation?
5. What happens if the coefficient of $x^2$ is positive instead of negative in the equation?

Tip:

In real-world applications, quadratic equations like this one are often used to model projectile motion, but always make sure to interpret your results in the context of the problem!