A snorkeler dives for a shell on a reef. After entering the​ water, the diver descends seven halves
7
2 ft in one second. Write an equation for a parabola that models the​ diver's vertical position y with respect to time x.

To model the snorkeler's vertical position \( y \) with respect to time \( x \), we can represent the diver's movement as a downward motion, meaning the equation will describe a parabola opening downwards.

Since the diver descends \( \frac{7}{2} \) feet per second, we can assume this is the initial rate of descent. We will use the general form of a parabola:

\[
y = ax^2 + bx + c
\]

Given the initial conditions:

1. **At \( x = 0 \) (starting point)**: The diver is at the surface of the water, so \( y(0) = 0 \). This gives \( c = 0 \).
2. **At \( x = 1 \) (after 1 second)**: The diver has descended \( \frac{7}{2} \) feet, so \( y(1) = -\frac{7}{2} \).

We can now substitute these values into the equation.

For \( x = 1 \) and \( y = -\frac{7}{2} \):

\[
-\frac{7}{2} = a(1)^2 + b(1)
\]
\[
-\frac{7}{2} = a + b
\]

At this point, we need more information or assumptions to fully determine both \( a \) and \( b \). If we assume the descent is primarily linear (i.e., we can ignore the acceleration effects), the equation simplifies to:

\[
y = -\frac{7}{2}x
\]

This assumes the vertical position is changing linearly with time. If the descent involves acceleration or curvature (as implied by a quadratic term), we need additional data, such as a second point on the curve, to solve for both \( a \) and \( b \).

Would you like to explore this further or provide more details for refinement?

### 5 Related Questions:
1. What would the equation look like if the diver accelerates while descending?
2. How can we find the time when the diver reaches a certain depth?
3. Can this model apply if the diver ascends back to the surface?
4. How does changing the initial descent rate affect the equation of the parabola?
5. What role does gravity play in modifying the descent of the diver?

### Tip:
When modeling vertical motion, it's important to consider whether the motion involves constant velocity (linear) or acceleration (quadratic).

A snorkeler dives for a shell on a reef. After entering the water, the diver descends seven halves (7/2) ft in one second. Write an equation for a parabola that models the diver's vertical position y with respect to time x.

Explore how to model a diver's vertical position using a quadratic equation, based on their descent rate of 7/2 ft per second. This problem illustrates key algebraic concepts such as parabolic and linear motion, suitable for students in Grades 9-11.

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