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We are given the equation of a parabola \( y^2 = 4(x + 1) \), and we need to find how fast the \(x\)-coordinate is changing at the given point when the \( y \)-coordinate is changing at a rate of 3 units per second.

### Step 1: Implicit differentiation
Start by differentiating the given equation with respect to time \( t \). The equation is:

\[
y^2 = 4(x + 1)
\]

Differentiating both sides with respect to \( t \), using the chain rule:

\[
2y \frac{dy}{dt} = 4 \frac{dx}{dt}
\]

### Step 2: Solve for \( \frac{dx}{dt} \)
Rearranging the equation to solve for \( \frac{dx}{dt} \):

\[
\frac{dx}{dt} = \frac{y \frac{dy}{dt}}{2}
\]

### Step 3: Substitute known values
The particle passes through the point \( (3, 16) \), so \( x = 3 \) and \( y = 16 \). We are also given that \( \frac{dy}{dt} = 3 \) units per second.

Substitute these values into the equation:

\[
\frac{dx}{dt} = \frac{16 \times 3}{2} = \frac{48}{2} = 24
\]

### Final Answer:
The rate at which the \( x \)-coordinate is changing is \( \frac{dx}{dt} = 24 \) units per second.

Would you like more details or have any questions?

Here are 5 questions related to this problem for further practice:
1. How would the rate of change of \( x \) vary if \( \frac{dy}{dt} \) were different?
2. What is the geometric interpretation of the relationship between \( \frac{dy}{dt} \) and \( \frac{dx}{dt} \)?
3. How does implicit differentiation apply to curves that are not functions?
4. How can this method be applied to curves in 3D, such as surfaces?
5. What happens if the point of interest on the curve is at the vertex of the parabola?

**Tip**: Always check that the point you are analyzing satisfies the given equation before performing differentiation.

A particle is moving along the parabola y^2 = 4(x + 1). As it passes through the point (3, 16), its y-coordinate is increasing at the rate of 3 units per second. How fast is the x-coordinate changing at this instant?

In this problem, a particle moves along the parabola y^2 = 4(x + 1). Given that the y-coordinate is increasing at 3 units per second, learn how to calculate the rate of change of the x-coordinate at the point (3, 16) using implicit differentiation and related rates.

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