h=215t -16t^2 , find all values of t for which the height is 97 feet

To find all values of \( t \) for which the height \( h \) is 97 feet, we'll substitute \( h = 97 \) into the equation and solve for \( t \):

\[
97 = 215t - 16t^2
\]

Rearrange the equation into standard form:

\[
-16t^2 + 215t - 97 = 0
\]

Now, we have a quadratic equation in the form \( at^2 + bt + c = 0 \), where:
- \( a = -16 \)
- \( b = 215 \)
- \( c = -97 \)

### Step 1: Solve Using the Quadratic Formula
The quadratic formula is:
\[
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substitute the values for \( a \), \( b \), and \( c \):

\[
t = \frac{-215 \pm \sqrt{(215)^2 - 4(-16)(-97)}}{2(-16)}
\]

### Step 2: Calculate the Discriminant
Calculate \( b^2 - 4ac \):
\[
(215)^2 - 4(-16)(-97) = 46225 - 6208 = 40017
\]

### Step 3: Solve for \( t \)
Substitute the values back into the quadratic formula:

\[
t = \frac{-215 \pm \sqrt{40017}}{-32}
\]

Now we can evaluate this further:

\[
t = \frac{-215 \pm 200.04}{-32}
\]

This gives two solutions for \( t \):
1. \[
t = \frac{-215 + 200.04}{-32} = \frac{-14.96}{-32} \approx 0.47
\]

2. \[
t = \frac{-215 - 200.04}{-32} = \frac{-415.04}{-32} \approx 12.97
\]

So, the values of \( t \) for which the height is 97 feet are approximately:
\[
t \approx 0.47 \text{ seconds} \quad \text{and} \quad t \approx 12.97 \text{ seconds}
\]

Would you like further details on any step, or do you have any questions?

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Here are some related questions that build on this problem:

1. What is the maximum height reached by the object?
2. For what values of \( t \) will the object be at half the maximum height?
3. How long does it take for the object to return to a height of zero?
4. Can we determine the object's velocity at \( t = 5 \) seconds?
5. How does changing the initial velocity affect the height at a given time?

**Tip:** The discriminant \( b^2 - 4ac \) tells you how many solutions a quadratic equation has: a positive value indicates two real solutions, zero means one real solution, and a negative value implies no real solutions.

h = 215t - 16t^2 , find all values of t for which the height is 97 feet.

Learn how to solve the quadratic equation h = 215t - 16t^2 to find the times when the height is 97 feet. The solution includes detailed steps using the quadratic formula and covers key algebra concepts. Suitable for students in Grades 9-11.

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