https://file.aiquickdraw.com/mathbot/file/1727923406420hi7p75nw.jpg

The problem involves a ball being thrown from a height of 217 feet with an initial downward velocity of 17 feet per second. The equation given for the height \( h \) of the ball after \( t \) seconds is:

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

We are tasked with finding how long it takes for the ball to hit the ground. This occurs when the height \( h = 0 \). So, we need to solve the equation:

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

### Step 1: Rearrange the equation
Rewriting the equation:

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

### Step 2: Solve using the quadratic formula
The quadratic formula is:

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

For our equation \( 16t^2 + 17t - 217 = 0 \), the coefficients are:
- \( a = 16 \)
- \( b = 17 \)
- \( c = -217 \)

Plugging these values into the quadratic formula:

\[
t = \frac{-17 \pm \sqrt{17^2 - 4(16)(-217)}}{2(16)}
\]
\[
t = \frac{-17 \pm \sqrt{289 + 13888}}{32}
\]
\[
t = \frac{-17 \pm \sqrt{14177}}{32}
\]
\[
t = \frac{-17 \pm 119.06}{32}
\]

### Step 3: Compute the two possible values for \( t \)
Now calculate the two possible solutions for \( t \):

1. \( t_1 = \frac{-17 + 119.06}{32} = \frac{102.06}{32} = 3.19 \)
2. \( t_2 = \frac{-17 - 119.06}{32} = \frac{-136.06}{32} = -4.25 \) (This solution is not physically meaningful because time cannot be negative.)

### Final Answer:
The ball hits the ground after approximately **3.19 seconds**.

Would you like a further explanation of any steps, or do you have any questions?

### Related Questions:
1. What is the significance of the negative root in quadratic equations like this one?
2. How would the equation change if the ball were thrown upward instead of downward?
3. How can we derive the time when the ball reaches its maximum height?
4. What would the velocity of the ball be when it hits the ground?
5. How can air resistance be incorporated into this model?

### Tip:
Always remember to check the context of a problem when solving quadratic equations, as one of the solutions may not make sense in real-world scenarios (e.g., negative time).

A ball is thrown from a height of 217 feet with an initial downward velocity of 17 ft/s. How long after the ball is thrown does it hit the ground?

Find out how long it takes for a ball thrown from a height of 217 feet with a downward velocity of 17 ft/s to hit the ground. Learn how to solve the quadratic equation for time using the quadratic formula. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation