The problem asks us to identify and sketch the graph of each of the given equations in three-space. I will focus on answering question 1. The equation is $4x^2 + 36y^2 = 144$.

1. Problem Description

The problem asks us to identify and sketch the graph of each of the given equations in three-space. I will focus on answering question

1. The equation is $4x^2 + 36y^2 = 144$.

2. Solution Steps

First, divide both sides of the equation by 144:

$\frac{4x^2}{144} + \frac{36y^2}{144} = \frac{144}{144}$

Simplify the fractions:

$\frac{x^2}{36} + \frac{y^2}{4} = 1$

This is an ellipse in the $xy$-plane. Since there is no $z$ term, the equation represents an elliptical cylinder in three-space. The ellipse is centered at the origin $(0,0,0)$. The semi-major axis is $a = \sqrt{36} = 6$ along the $x$-axis, and the semi-minor axis is $b = \sqrt{4} = 2$ along the $y$-axis. The cylinder extends infinitely along the $z$-axis.

3. Final Answer

The equation $4x^2 + 36y^2 = 144$ represents an elliptical cylinder whose axis is the $z$-axis. The cross-section in the $xy$-plane is an ellipse with semi-major axis 6 along the $x$-axis and semi-minor axis 2 along the $y$-axis.