Test #2 MATH 271 NAME: --Please show all your own work to be eligible for partial credit. 1.) The temperature X aBß Cb anywhere on a 2-dimensional sheet of metal is the distance from a point on the sheet to the origin on the sheet. Show that the direction of greatest change in the temperature of the solid is directly towards or directly away from the origin on the plate.
2.) The temperature X aBß Cß D b anywhere contained within a 3-dimensional solid of metal is the distance from a point in the solid to the origin in the solid. Show that the direction of greatest change in the temperature of the solid is directly towards or directly away from the origin in the solid. 3.) An industrial tank must be designed in the shape of a cylinder with height 2 and radius <, and must also have hemispherical ends. If the container must hold a total of 500 liters of fluid, determine the radius < and length 2 that minimizes the amount of material used in the construction of the tank. 4.) A heated room is shaped of a rectangular box and has a volume of 1500 square feet. Due to the behavior of warm air the heat loss in the room is 5 times greater through the ceiling of the room when compared to the floor. As for the walls of the room, the heat loss is three times greater through them when compared to the floor. Determine the dimensions of the room that will minimize heat loss and therefore minimize heating costs. 5.) Find parametric equations of the line tangent to the surface D œ C# B$ C at the point a#ß "ß *b Whose projection onto the BC-plane is..... a.) Parallel to the B-axis. b.) Parallel to the C-axis c.) Parallel to line C œ B 6.) The part of a tree normally sawed into lumber is the trunk, a solid shaped approximately like a right circular cylinder. If the radius of the trunk for a certain tree is growing at a rate of "# inch per year and the height is increasing at a rate of 8 inches per year, how fast is the volume of the tree increasing by when the radius is 20 inches and the height is 400 inches? Express your answer in board-feet. One board foot is equivalent to a volume of space 1 inch by 12 inches by 12 inches.
7.) Sand is pouring onto a conical pile in such a way at a certain instance when the pile is 100 inches in height and increasing at a rate of 3 inches per minute, the base has a radius of 40 inches and is increasing at a rate of 2 inches per minute. At this instance how is the volume increasing?
8.) Consider the curve of intersection for the two surfaces 0 aBß Cß D b œ *B# %C# %D # %" œ ! and 1aBß Cß D b œ #B# C# $D # "! œ !Þ Find the parametric equations for the line tangent to the curve representing this intersection at the point T a"ß #ß #b. 9.) Find the volume of the solid in the first octant bounded by the surface *B# %C# œ $' and the plane *B %C 'D œ !Þ "!ÞÑ Find the volume of the solid in the first octant bounded by the circular cylinders B# C# œ "' and C# D # œ "', and the coordinate planes. 11.) Using iterated integration find the area of the shaded region V illustrated as follows.
"#ÞÑ Find the area for one leaf of the four leaf rose curve given by the equation < œ +=38Ð#)ÑÞ "$ÞÑ Find the area for the region outside the circle < œ # and inside the lemniscate <# œ *-9=a#)bÞ "%ÞÑ Find the mass and center or mass for the planar lamina bounded by the given curves, C œ /B ß C œ !ß B œ !, B œ " and has a density of 3aBß Cb œ # B CÞ "&ÞÑ Find the moments of inertia for the planar lamina bounded by the given curves C œ B# , C œ % and has a density of 3aBß Cb œ CÞ