91Ó°ÊÓ

Describe the solid satisfying the condition. \(x^{2}+y^{2}+z^{2}>4\)

An object is pulled 10 feet across a floor, using a force of 85 pounds. The direction of the force is \(60^{\circ}\) above the horizontal. Find the work done.

A toy wagon is pulled by exerting a force of 25 pounds on a handle that makes a \(20^{\circ}\) angle with the horizontal. Find the work done in pulling the wagon 50 feet.

Find an equation of the plane. The plane contains the \(y\) -axis and makes an angle of \(\pi / 6\) with the positive \(x\) -axis

The vector \(v\) and its initial point are given. Find the terminal point. \(\mathbf{v}=\left\langle 1,-\frac{2}{3}, \frac{1}{2}\right\rangle\) Initial point: \(\left(0,2, \frac{5}{2}\right)\)

Geography Because of the forces caused by its rotation, Earth is an oblate ellipsoid rather than a sphere. The equatorial radius is 3963 miles and the polar radius is 3950 miles. Find an equation of the ellipsoid. (Assume that the center of Earth is at the origin and that the trace formed by the plane \(z=0\) corresponds to the equator.)

Consider a regular tetrahedron with vertices \((0,0,0),(k, k, 0),(k, 0, k),\) and \((0, k, k),\) where \(k\) is a positive real number. (a) Sketch the graph of the tetrahedron. (b) Find the length of each edge. (c) Find the angle between any two edges. (d) Find the angle between the line segments from the centroid \((k / 2, k / 2, k / 2)\) to two vertices. This is the bond angle for a molecule such as \(\mathrm{CH}_{4}\) or \(\mathrm{PbCl}_{4}\), where the structure of the molecule is a tetrahedron.

Consider the vectors \(\mathbf{u}=\langle\cos \alpha, \sin \alpha, 0\rangle\) and \(\mathbf{v}=\langle\cos \beta, \sin \beta, 0\rangle\) where \(\alpha>\beta\) Find the dot product of the vectors and use the result to prove the identity \(\cos (\alpha-\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta\).

Spiral of Archimedes The curve represented by the equation \(r=a \theta,\) where \(a\) is a constant, is called the spiral of Archimedes. (a) Use a graphing utility to graph \(r=\theta,\) where \(\theta \geq 0\). What happens to the graph of \(r=a \theta\) as \(a\) increases? What happens if \(\theta \leq 0 ?\) (b) Determine the points on the spiral \(r=a \theta(a>0, \theta \geq 0)\) where the curve crosses the polar axis. (c) Find the length of \(r=\theta\) over the interval \(0 \leq \theta \leq 2 \pi\). (d) Find the area under the curve \(r=\theta\) for \(0 \leq \theta \leq 2 \pi\)

Use vectors to determine whether the points are collinear. (0,0,0),(1,3,-2),(2,-6,4)