91Ó°ÊÓ

find the distance from the point to the plane. $$(1,0,-1), \quad-4 x+y+z=4$$

Describe the given set with a single equation or with a pair of equations. The set of points in space that lie 2 units from the point (0,0,1) and, at the same time, 2 units from the point (0,0,-1)

a. Express the area \(A\) of the cross-section cut from the ellipsoid $$x^{2}+\frac{y^{2}}{4}+\frac{z^{2}}{9}=1$$ by the plane \(z=c\) as a function of \(c .\) (The area of an ellipse with semiaxes \(a\) and \(b\) is \(\pi a b .\) ) b. Use slices perpendicular to the \(z\) -axis to find the volume of the ellipsoid in part (a). c. Now find the volume of the ellipsoid $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 $$ Does your formula give the volume of a sphere of radius \(a\) if \(a=b=c ?\)

Find the areas of the triangles whose vertices are given. $$A(1,0,0), \quad B(0,2,0), \quad C(0,0,-1)$$

Find the distance from the plane \(x+2 y+6 z=1\) to the plane \(x+2 y+6 z=10\)

Write inequalities to describe the sets The slab bounded by the planes \(z=0\) and \(z=1\) (planes included)

When solving you may need to use a calculator or a computer. Velocity \(\quad\) An airplane is flying in the direction \(25^{\circ}\) west of north at \(800 \mathrm{km} / \mathrm{h} .\) Find the component form of the velocity of the airplane, assuming that the positive \(x\) -axis represents due east and the positive \(y\) -axis represents due north.

Find the areas of the triangles whose vertices are given. $$A(0,0,0), \quad B(-1,1,-1), \quad C(3,0,3)$$

Write inequalities to describe the sets. The solid cube in the first octant bounded by the coordinate planes and the planes \(x=2, y=2,\) and \(z=2\)

Find the distance from the line \(x=2+t, y=1+t\) \(z=-(1 / 2)-(1 / 2) t\) to the plane \(x+2 y+6 z=10\)