To determine the points at which the curve $\mathbf{r}(t) = 3t \mathbf{i} + (5t - t^2) \mathbf{k}$ intersects the paraboloid given by the equation $z = x^2 + 3y^2$, we need to match the parametric form of the curve to the equation of the paraboloid.

Step 1: Identify the components of the curve

From the parametric equation of the curve $\mathbf{r}(t) = 3t \mathbf{i} + (5t - t^2) \mathbf{k}$, we can identify:

• $x(t) = 3t$ (from the $i$-component)
• $y(t) = 0$ (there is no $j$-component, so $y = 0$)
• $z(t) = 5t - t^2$ (from the $k$-component)

Step 2: Substitute $x(t)$ and $y(t)$ into the equation of the paraboloid

The equation of the paraboloid is: $z = x^2 + 3y^2$

Substitute $x = 3t$ and $y = 0$ into this equation: $z = (3t)^2 + 3(0)^2 = 9t^2$

Step 3: Set $z$ from the curve equal to $z$ from the paraboloid

Now, equate the expression for $z(t)$ from the curve to the expression for $z$ from the paraboloid: $5t - t^2 = 9t^2$

Step 4: Solve for $t$

Rearrange the equation: $5t - t^2 = 9t^2$ $5t = 10t^2$ $t(5 - 10t) = 0$

So, the solutions are: $t = 0 \quad \text{or} \quad t = \frac{1}{2}$

Step 5: Find the points of intersection

Now, we substitute these values of $t$ back into the parametric equations for $x$, $y$, and $z$.

• When $t = 0$: $x = 3(0) = 0, \quad y = 0, \quad z = 5(0) - 0^2 = 0$ So, one point of intersection is $(0, 0, 0)$.

• When $t = \frac{1}{2}$: $x = 3\left( \frac{1}{2} \right) = \frac{3}{2}, \quad y = 0, \quad z = 5\left( \frac{1}{2} \right) - \left( \frac{1}{2} \right)^2 = \frac{5}{2} - \frac{1}{4} = \frac{9}{4}$ So, the second point of intersection is $\left( \frac{3}{2}, 0, \frac{9}{4} \right)$.

Final Answer:

The curve intersects the paraboloid at the following points:

• $(0, 0, 0)$
• $\left( \frac{3}{2}, 0, \frac{9}{4} \right)$

Would you like more details on any step, or have any questions?

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Here are five related questions for further practice:

1. What are the parametric equations for the intersection of two surfaces?
2. How can you determine if a curve lies entirely on a surface?
3. What is the geometrical significance of a parametric curve and its projection on a surface?
4. How do we find the tangent line to the curve at the point of intersection with the paraboloid?
5. Can a curve intersect a surface at more than two points?

Tip: When solving for intersections, always make sure to check for multiple solutions, as parametric equations often have more than one solution.