We are given a point $(7, 3, 8)$ and a parameterized curve described by:

$x(t) = 2t + 7, \quad y(t) = 4t^{3/2} + 3, \quad z(t) = \frac{t^2}{2} + 8$

We need to find where we end up after moving 3 units along the curve in the positive direction.

Step 1: Parametrize the curve.

The velocity vector along the curve is the derivative of the position functions $x(t), y(t), z(t)$ with respect to $t$:

$\mathbf{v}(t) = \left( \frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt} \right)$

So, we differentiate each component:

$\frac{dx}{dt} = 2, \quad \frac{dy}{dt} = 6t^{1/2}, \quad \frac{dz}{dt} = t$

Thus, the velocity vector is:

$\mathbf{v}(t) = \langle 2, 6t^{1/2}, t \rangle$

Step 2: Calculate the speed.

The speed is the magnitude of the velocity vector:

$\|\mathbf{v}(t)\| = \sqrt{(2)^2 + (6t^{1/2})^2 + (t)^2} = \sqrt{4 + 36t + t^2}$

Step 3: Arc length formula.

The distance traveled along a curve is given by the arc length formula:

$s(t) = \int_0^t \|\mathbf{v}(\tau)\| \, d\tau = \int_0^t \sqrt{4 + 36\tau + \tau^2} \, d\tau$

We need to find $t$ such that the arc length equals 3 units, i.e.,

$\int_0^t \sqrt{4 + 36\tau + \tau^2} \, d\tau = 3$

Solving this integral directly can be complex, so let me approximate this numerically to find the value of $t$ corresponding to 3 units of movement.The value of $t$ that corresponds to moving 3 units along the curve is approximately $t \approx 0.739$.

Step 4: Calculate the new position.

Now, using $t = 0.739$, we can find the new position by plugging this value into the parametric equations for $x(t), y(t), z(t)$:

$x(0.739) = 2(0.739) + 7 = 8.478$ $y(0.739) = 4(0.739)^{3/2} + 3 \approx 4.957$ $z(0.739) = \frac{(0.739)^2}{2} + 8 \approx 8.273$

Thus, after moving 3 units along the curve, your new position is approximately:

$(8.478, 4.957, 8.273)$

Would you like any further details or explanations?

Here are 5 related questions to expand on this topic:

1. How can you approximate arc length for curves with complex parameterizations?
2. How does the velocity vector help in understanding movement along a curve?
3. What are some techniques for solving integrals involving arc length?
4. How does curvature of a curve affect arc length?
5. What other methods can be used to solve this kind of parametric motion problem?

Tip: Numerical methods like integration and root-finding are often used when exact solutions to arc length problems are difficult to obtain.