A ball is thrown from a height of 217 feet with an initial downward velocity of 17 ft/s. The ball's height h (in feet) after t seconds is given by the following. h= 217 - 17t - 16t^2 How long after the ball is thrown does it hit the ground? Round your answers) to the nearest hundredth. (If there is more than one answer, use the "or" button.).


To find when the ball hits the ground, we need to determine when the height \( h \) is equal to zero. The given equation for height is:

\[
h(t) = 217 - 17t - 16t^2
\]

Setting \( h(t) = 0 \) to represent the ball hitting the ground, we get:

\[
0 = 217 - 17t - 16t^2
\]

Rearranging the equation:

\[
16t^2 + 17t - 217 = 0
\]

Now, we solve this quadratic equation using the quadratic formula:

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

Where \( a = 16 \), \( b = 17 \), and \( c = -217 \). Substituting these values into the quadratic formula:

\[
t = \frac{-17 \pm \sqrt{17^2 - 4(16)(-217)}}{2(16)}
\]

\[
t = \frac{-17 \pm \sqrt{289 + 13888}}{32}
\]

\[
t = \frac{-17 \pm \sqrt{14177}}{32}
\]

Let's calculate the value of \( t \).The two solutions for \( t \) are:

\[
t_1 = 3.19 \quad \text{seconds}
\]
\[
t_2 = -4.25 \quad \text{seconds (not physically meaningful since time cannot be negative)}
\]

Thus, the ball hits the ground after **3.19 seconds**.

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you solve quadratic equations in general?
2. What happens if the initial velocity is upward instead of downward?
3. How would the height equation change if gravity was different?
4. Can we graph the height function to visualize the ball's path?
5. What is the significance of the discriminant in quadratic equations?

**Tip:** In quadratic equations, negative time values are usually ignored in real-world applications where time must be positive.

A ball is thrown from a height of 217 feet with an initial downward velocity of 17 ft/s. The ball's height h (in feet) after t seconds is given by the following. h = 217 - 17t - 16t^2. How long after the ball is thrown does it hit the ground? Round your answer to the nearest hundredth.

Learn how to solve a quadratic equation to find when a ball thrown from a height of 217 feet hits the ground. This problem explores key concepts in algebra and projectile motion, using the quadratic formula to calculate the time. Suitable for students in Grades 9-12.

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