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c Math 151 WIR, Spring 2014, Benjamin Aurispa Math 151 Week in Review 12 Sections 5.5, 5.7 1. A farmer wants to fence in a rectangular area along a river and divide it into 3 equal-sized pens with fencing perpendicular to the water. He is not going to build a fence along the river. If he has 120 ft of fencing available to use, find the maximum area of the field. 2. Find the closest point on the line y = −3x + 4 to the point (1, 5). 3. A box with an open top is to be constucted from 30 m2 of cardboard. The length of the box is to be twice its width. Find the dimensions of the box that will maximize the volume. 4. A company is designing a cylindrical can which will have a volume of 2000 cm3 . Find the dimensions of the can that would minimize the surface area. 5. Find the dimensions of the largest rectangle that can be inscribed in the area bounded by the top half of the ellipse 4x2 + y 2 = 64 and the x-axis. What is the maximum area? 6. A 20-m wire is cut into two pieces. One piece is bent into an equilateral triangle and the other is bent into a square. How should the wire be cut (if at all) so that the total area enclosed is a maximum? minimum? 7. A box with an open top is to be constructed from a piece of cardboard that is 4 ft × 8 ft by cutting out a square from each of the four corners and folding up the sides. What size square should be cut out to attain the largest possible volume? 8. Find f (x) for the following. (a) (b) (c) (d) 3 1 =x x − 6 + x x 1 + cos x + ex , f (0) = 7 f 0 (x) = 5 + 2 x +1 √ x + x5/4 00 f (x) = , f 0 (1) = 5, f (0) = 4 x f 00 (x) = x − 4 − ex , f (0) = 3, f (1) = 2 f 0 (x) 2 2 9. An object is thrown down from a 500 ft ledge with a speed of 15 ft/s. (a) Find the distance of the object above the ground at time t. (b) When and with what velocity does the object hit the ground? 10. A car traveling at a speed of 30 mph (44 ft/s) applies a constant acceleration of 8 ft/s2 . How far will the car travel before reaching a speed of 60 mph (88 ft/s)? 11. An object in motion has acceleration given by a(t) =< sin t + t3 + 1, et + 4 >. Find the position vector function given that v(0) =< 0, 5 > and r(0) =< 3, 0 >. 1