https://file.aiquickdraw.com/mathbot/file/1725642730906x04bg0j2.jpg

It looks like the image contains a problem about the path of a diver. Let me transcribe and solve it:

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**Problem Transcription:**
"The path of a diver is given by the equation:
\[ y = -0.7x^2 + 7.5x \]
where \( y \) represents the height in meters above the water, and \( x \) represents the horizontal distance in meters from the diving board. The diver weighs 70 kg and is weighted by the water. What is the maximum height the diver is above the water?"

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### Solution:
We need to find the maximum height of the diver. The equation that models the diver's path is a quadratic equation:
\[
y = -0.7x^2 + 7.5x
\]
This equation is a parabola, and since the coefficient of \(x^2\) is negative, the parabola opens downwards, indicating that the vertex represents the maximum point.

To find the maximum height, we first need to determine the vertex of the parabola.

#### Step 1: Find the x-coordinate of the vertex
The formula for the x-coordinate of the vertex of a parabola \( y = ax^2 + bx + c \) is given by:
\[
x_{\text{vertex}} = \frac{-b}{2a}
\]
Here, \( a = -0.7 \) and \( b = 7.5 \). Plugging these values into the formula:

\[
x_{\text{vertex}} = \frac{-7.5}{2(-0.7)} = \frac{-7.5}{-1.4} \approx 5.36
\]

#### Step 2: Find the corresponding y-coordinate
Now, substitute \( x = 5.36 \) back into the equation to find the maximum height:

\[
y = -0.7(5.36)^2 + 7.5(5.36)
\]

Let me calculate this.The horizontal distance \(x_{\text{vertex}}\) at which the diver reaches the maximum height is approximately 5.36 meters. The maximum height \(y_{\text{max}}\) is approximately 20.09 meters above the water.

Let me know if you'd like more details, or if you have any questions!

Here are some related questions:
1. How does the quadratic formula help determine the maximum height?
2. What is the physical significance of the coefficients in the diver's path equation?
3. How would the diver's weight affect the path in reality (if not neglecting forces)?
4. Can the same method be used to find the minimum in a different quadratic equation?
5. What would happen to the maximum height if the initial velocity changed?

**Tip**: The vertex form of a quadratic equation is often useful for understanding the peak of a parabola without completing the square.

The path of a diver is given by the equation: y = -0.7x^2 + 7.5x, where y represents the height in meters above the water, and x represents the horizontal distance in meters from the diving board. The diver weighs 70 kg and is weighted by the water. What is the maximum height the diver is above the water?

Discover how to find the maximum height of a diver above the water using the quadratic equation y = -0.7x^2 + 7.5x. This problem involves key algebraic concepts such as the vertex of a parabola and the quadratic equation, suitable for students in Grades 9-11.

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