) At what points does the curve r(t)=3ti+(5t−t^2)k
 intersect the paraboloid z=x^2+3y^2

when t=
at the point(s)=

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.

At what points does the curve r(t)=3ti+(5t−t^2)k intersect the paraboloid z=x^2+3y^2?

Learn how to determine the points where the parametric curve r(t) = 3ti + (5t − t^2)k intersects the paraboloid z = x^2 + 3y^2. This problem involves solving a quadratic equation derived from parametric equations. Suitable for undergraduate calculus students.

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