8—Multivariable Calculus 2478.44 What is the shortest distance from the origin to the plane defined byA~.(~r−~r 0 ) = 0? Do this using
Lagrange multipliers, and then explain why of course the answer is correct.
8.45 The U.S. Post Office has decided to use a norm like Eq. (6.7) 2 to measure boxes. The size is defined
to be the sum of the height and the circumference of the rectangular box, and the circumference is around the
thickest part of the package. What is the maximum volume you can ship if this size is constrained to be less than
130 inches? Does it matter if the box is rectangular or cylindrical? SeeUSPS
8.46 Plotθversusbin equation ( 31 ) or ( 32 ).
8.47 A disk of radiusRis at a distancecabove thex-yplane and parallel to that plane. What is the solid angle
that this disk subtends from the origin?
8.48 Within a sphere of radiusR, what is the volume contained between the planes defined byz=aandz=b?
8.49 Find the mean-square distance,V^1
∫
r^2 dV, from a point on the surface of a sphere to points inside the
sphere. Note: Plan ahead and try to make this problem as easy as possible.
8.50 Find the mean distance,V^1
∫
r dV, from a point on the surface of a sphere to points inside the sphere.8.51 A volume mass density is specified in spherical coordinates to be
ρ(r,θ,φ) =ρ 0(
1 +r^2 /R^2)[
1 +^1 / 2 cosθsin^2 φ+^1 / 4 cos^2 θsin^3 φ]
Compute the total mass in the volume 0 < r < R.