Two water balloons were launched into the air at different moments and collided. The water balloons were modeled by the quadratic functions: y = −3x2 + 24x + 3 and y = −4x2 + 25x + 15, where y represents the height in meters and x represents the time in seconds after the launch. What is the time, in seconds, that the balloons collided at the highest point?

To find the time when the water balloons collided at the highest point, we need to determine the point of collision for the two given quadratic functions:

1. \( y = -3x^2 + 24x + 3 \)
2. \( y = -4x^2 + 25x + 15 \)

First, set the equations equal to each other to find the \( x \) values where the collision occurs:
\[ -3x^2 + 24x + 3 = -4x^2 + 25x + 15 \]

Rearrange the equation to set it to zero:
\[ -3x^2 + 24x + 3 + 4x^2 - 25x - 15 = 0 \]
\[ x^2 - x - 12 = 0 \]

Next, solve the quadratic equation \( x^2 - x - 12 = 0 \). This can be factored:
\[ (x - 4)(x + 3) = 0 \]

So, the solutions for \( x \) are:
\[ x = 4 \quad \text{and} \quad x = -3 \]

Since time cannot be negative, we discard \( x = -3 \) and accept \( x = 4 \).

Now, we need to determine the height at \( x = 4 \) to find the highest point of the collision. Substitute \( x = 4 \) into both quadratic equations to find \( y \):

For \( y = -3x^2 + 24x + 3 \):
\[ y = -3(4)^2 + 24(4) + 3 \]
\[ y = -48 + 96 + 3 \]
\[ y = 51 \]

For \( y = -4x^2 + 25x + 15 \):
\[ y = -4(4)^2 + 25(4) + 15 \]
\[ y = -64 + 100 + 15 \]
\[ y = 51 \]

Both equations yield the same height \( y = 51 \) meters at \( x = 4 \) seconds, confirming that the collision occurs at this point.

Therefore, the time at which the balloons collided at the highest point is \( \boxed{4 \text{ seconds}} \).

Do you want more details or have any questions?

Here are 5 additional questions for you to consider:
1. How can you determine the vertex of a quadratic function?
2. What are the methods to solve a quadratic equation?
3. How do you find the points of intersection between two functions?
4. Can quadratic functions have more than one point of intersection?
5. How do the coefficients of a quadratic equation affect its graph?

**Tip:** When solving quadratic equations, always check if the solutions make sense in the context of the problem, especially if the variable represents a physical quantity like time.

Explore how to find the time when two water balloons collide at their highest point, modeled by quadratic functions. Learn the step-by-step solution involving quadratic equations and their applications. Suitable for high school students.

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