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c Math 151 WIR, Fall 2013, Benjamin Aurispa Math 151 Week in Review 12 Sections 5.5, 5.7, 6.1 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). 1 c Math 151 WIR, Fall 2013, Benjamin Aurispa 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. 2 c Math 151 WIR, Fall 2013, Benjamin Aurispa 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. 3 c Math 151 WIR, Fall 2013, Benjamin Aurispa 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? 4 c Math 151 WIR, Fall 2013, Benjamin Aurispa 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? 5 c Math 151 WIR, Fall 2013, Benjamin Aurispa 7. Find f (x) for the following. (a) f ′ (x) =x 2 3 x − 6 x (b) f ′ (x) = 5 + (c) f ′′ (x) = √ 2 + 1 x 1 + cos x + ex , f (0) = 7 x2 + 1 x + x5/4 , f ′ (1) = 5, f (0) = 4 x (d) f ′′ (x) = x − 4 − ex , f (0) = 3, f (1) = 2 6 c Math 151 WIR, Fall 2013, Benjamin Aurispa 8. 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? 9. 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)? 7 c Math 151 WIR, Fall 2013, Benjamin Aurispa 10. 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 >. 11. Calculate the following sums. (a) (b) 4 X i=1 (2i2 − 3i) 35 X (3i − 3i+1 ) i=5 8 c Math 151 WIR, Fall 2013, Benjamin Aurispa (c) 20 X (i3 + 4) n X (3i + 2)2 i=3 (d) i=1 12. Represent the following sum using sigma notation. 13. Calculate lim n→∞ n X 2 2i 3 i=1 n n 9 400 4 9 16 25 + + + + ··· + 7 8 9 10 25