The height of a ball dropped from a 100-foot platform is represented
by the function: h(t) = 100 - 16t?, where t is the time in seconds since the ball was dropped.
Write a function that gives the time it will take the ball to reach a height of h.

The given function for the height of the ball is:

\[
h(t) = 100 - 16t^2
\]

Here, \( t \) represents the time in seconds, and \( h(t) \) is the height of the ball at time \( t \).

To find the time it will take for the ball to reach a specific height \( h \), we need to solve for \( t \) in terms of \( h \). Let's rewrite the equation to express \( t \) as a function of \( h \).

### Step 1: Set up the equation for \( h \):

\[
h = 100 - 16t^2
\]

### Step 2: Solve for \( t^2 \):

Subtract 100 from both sides:

\[
h - 100 = -16t^2
\]

Now, divide both sides by -16:

\[
t^2 = \frac{100 - h}{16}
\]

### Step 3: Solve for \( t \):

Take the square root of both sides:

\[
t = \pm \sqrt{\frac{100 - h}{16}}
\]

Since time \( t \) must be non-negative, we only take the positive root:

\[
t = \sqrt{\frac{100 - h}{16}}
\]

### Final Function:

The function that gives the time \( t \) when the ball reaches a height \( h \) is:

\[
t(h) = \sqrt{\frac{100 - h}{16}}
\]

Would you like further details or examples for how to use this function?

Here are 5 related questions to think about:
1. How long will it take for the ball to reach a height of 50 feet?
2. What is the maximum height the ball can reach after being dropped?
3. At what time will the ball hit the ground (i.e., \( h = 0 \))?
4. What happens to the function if the ball is dropped from a different height?
5. How would the equation change if gravity had a different constant?

**Tip:** When solving quadratic equations, always check if taking the square root introduces any positive-negative ambiguity in the solution!

The height of a ball dropped from a 100-foot platform is represented by the function: h(t) = 100 - 16t², where t is the time in seconds since the ball was dropped. Write a function that gives the time it will take the ball to reach a height of h.

Explore how to find the time it takes for a ball dropped from 100 feet to reach a given height using quadratic equations. This problem involves algebraic concepts like solving for t in terms of h using the formula h(t) = 100 - 16t². Suitable for Grades 9-10.

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