Methane, \(CH_4\), is a molecule with one carbon and four hydrogen atoms. Assume the hydrogen atoms are located at the corners of a regular tetrahedron with coordinates (0,0,0), (1,0,1), (0,1,1), and (1,1,0). The carbon atom is at the centroid of this tetrahedron at (1/2, 1/2, 1/2). The bond angle for the molecule is the angle between the line segments joining the carbon atom to two of the hydrogen atoms. Show this angle is approximately 109.5\(^\circ\).
Consider the triangle with vertices at (0,1,2), (4,3,3) and (3,4,5). Which statement below is most accurate?
Show that \(x = a \cos(t) + h\) and \( y = b \sin(t) + k\) are parametric equations for an ellipse with center at \(h,k\) and axes lengths \(2a\) and \(2b\).
Find the equation for the tangent to the curve $$ x = \sec t \hspace{1cm} y = \tan t \hspace{1cm} -\pi/2 < t < \pi/2$$ at the point \(t = \pi/4\).
Find the area of one leaf of the three leaf rose \(r = 2 \cos (3\theta)\).
Sketch the graph of the curve \(r \cos \theta = 2 r \sin \theta \).
Find the area of the astroid with parametric equations below. (hint: exploit symmetry)
$$x = \cos^3 t,\hspace{1cm} y=\sin^3 t, \hspace{1cm} 0 \le t \le 2\pi$$
