Use traces to sketch the surface. (If an answer does not exist, enter DNE. Select Update Graph to see your response plotted on the screen. Select the Submit button to grade your response.) $x^2 = 64y^2 + z^2$

Find:

• (i) (Write an equation for the cross section at $z = -8$ using $x$ and $y$.)

• (ii) (Write an equation for the cross section at $z = 0$ using $x$ and $y$.)

• (iii) (Write an equation for the cross section at $z = 8$ using $x$ and $y$.)

• (iv) (Write an equation for the cross section at $y = -8$ using $x$ and $z$).

• (v) (Write an equation for the cross section at $y = 0$ using $x$ and $z$).

• (vi) (Write an equation for the cross section at $y = 8$ using $x$ and $z$).

• (vii) (Write an equation for the cross section at $x = -8$ using $y$ and $z$).

• (viii) (Write an equation for the cross section at $x = 8$ using $y$ and $z$).

Identify the surface

• ellipsoid
• elliptic paraboloid
• elliptic cylinder
• hyperbolic paraboloid
• hyperboloid of two sheets
• parabolic cylinder
• elliptic cone
• hyperboloid of one sheet

Neetesh Kumar | December 15, 2024

Calculus Homework Help

This is the solution to Math 1C
Assignment: 12.6 Question Number 9
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Step-by-step solution:

Step 1: Analyze the given equation

The equation of the surface is: $x^2 = 64y^2 + z^2.$

This represents a cone since it is quadratic in $x$, $y$, and $z$, and there is no linear term.

Step 2: Find the cross sections

(i) Cross section at $z = -8$

Substitute $z = -8$ into $x^2 = 64y^2 + z^2$: $x^2 = 64y^2 + (-8)^2$

Simplify: $\boxed{x^2 = 64y^2 + 64}$

This is an equation of a hyperbola in the $xy$-plane.

(ii) Cross section at $z = 0$

Substitute $z = 0$ into $x^2 = 64y^2 + z^2$: $\boxed{x^2 = 64y^2}$

Simplify: $x^2 - 64y^2 = 0.$

This is a pair of intersecting lines in the $xy$-plane.

(iii) Cross section at $z = 8$

Substitute $z = 8$ into $x^2 = 64y^2 + z^2$: $x^2 = 64y^2 + (8)^2.$

Simplify: $\boxed{x^2 = 64y^2 + 64}$

This is an equation of a hyperbola in the $xy$-plane.

(iv) Cross section at $y = -8$

Substitute $y = -8$ into $x^2 = 64y^2 + z^2$: $x^2 = 64(-8)^2 + z^2.$

Simplify: $x^2 = 64(64) + z^2.$

$\boxed{x^2 = 4096 + z^2}$

This is an equation of a hyperbola in the $xz$-plane.

(v) Cross section at $y = 0$

Substitute $y = 0$ into $x^2 = 64y^2 + z^2$: $\boxed{x^2 = z^2}$

Simplify: $x^2 - z^2 = 0.$

This is a pair of intersecting lines in the $xz$-plane.

(vi) Cross section at $y = 8$

Substitute $y = 8$ into $x^2 = 64y^2 + z^2$: $x^2 = 64(8)^2 + z^2.$

Simplify: $\boxed{x^2 = 4096 + z^2}$

This is an equation of a hyperbola in the $xz$-plane.

(vii) Cross section at $x = -8$

Substitute $x = -8$ into $x^2 = 64y^2 + z^2$: $(-8)^2 = 64y^2 + z^2.$

Simplify: $\boxed{64 = 64y^2 + z^2}$

Rearrange: $y^2 + \frac{z^2}{64} = 1.$

This is an equation of an ellipse in the $yz$-plane.

(viii) Cross section at $x = 8$

Substitute $x = 8$ into $x^2 = 64y^2 + z^2$: $(8)^2 = 64y^2 + z^2.$

Simplify: $\boxed{64 = 64y^2 + z^2}$

Rearrange: $y^2 + \frac{z^2}{64} = 1.$

This is an equation of an ellipse in the $yz$-plane.

Step 3: Identify the surface

The surface $x^2 = 64y^2 + z^2$ is an $\boxed{\text{elliptic cone}}$.

Final Answer:

(i) Cross section at $z = -8$

$\boxed{x^2 = 64y^2 + 64}$

(ii) Cross section at $z = 0$

$\boxed{x^2 = 64y^2}$

(iii) Cross section at $z = 8$

$\boxed{x^2 = 64y^2 + 64}$

(iv) Cross section at $y = -8$

$\boxed{x^2 = 4096 + z^2}$

(v) Cross section at $y = 0$

$\boxed{x^2 = z^2}$

(vi) Cross section at $y = 8$

$\boxed{x^2 = 4096 + z^2}$

(vii) Cross section at $x = -8$

$\boxed{64 = 64y^2 + z^2}$

(viii) Cross section at $x = 8$

$\boxed{64 = 64y^2 + z^2}$

Identify the surface

$\boxed{\text{elliptic cone}}$

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