Identify the surface with the given vector equation. $\mathbf{r}(s, t) = \langle s \sin(7t), s^2, s \cos(7t) \rangle$

• elliptic cylinder
• circular cylinder
• plane
• hyperbolic paraboloid
• circular paraboloid

Neetesh Kumar | November 14, 2024

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This is the solution to Math 1D
Assignment: 16.6 Question Number 6
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Step-by-step solution:

To determine the surface type, let’s analyze each component of the given vector equation: $\mathbf{r}(s, t) = \langle s \sin(7t), s^2, s \cos(7t) \rangle$

This vector equation gives the following coordinates:

• For the x-coordinate: $x = s \sin(7t)$
• For the y-coordinate: $y = s^2$
• For the z-coordinate: $z = s \cos(7t)$

Step 1: Analyze the Relationship Between $x$, $y$, and $z$

Notice that both $x$ and $z$ contain $s$ multiplied by trigonometric functions of $t$. To find a relationship between $x$, $y$, and $z$, let’s square the $x$ and $z$ components and add them: $x^2 = s^2 \sin^2(7t)$ $z^2 = s^2 \cos^2(7t)$

Adding these two equations: $x^2 + z^2 = s^2 (\sin^2(7t) + \cos^2(7t))$

Using the identity $\sin^2(7t) + \cos^2(7t) = 1$, we get: $x^2 + z^2 = s^2$

Since $y = s^2$, we can substitute $y$ for $s^2$: $x^2 + z^2 = y$

Step 2: Identify the Surface Type

The equation $x^2 + z^2 = y$ represents a circular paraboloid. This is the standard form of a circular paraboloid, opening along the $y$-axis.

Answer:

The correct option is E: circular paraboloid

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