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c Math 151 WIR, Spring 2010, 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). 3. A 20-m wire is cut into two pieces. One piece is bent into an equilateral triangle and the other is bent into a circle. How should the wire be cut (if at all) so that the total area enclosed is a maximum? minimum? 4. A company is designing a cylinder which will have a volume of 2000 cm3 . Find the dimensions of the cylinder that would minimize the cost of production. 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. 6. Find f (x) for the following. 3 1 + 4 x x 1 + cos x + ex , f (0) = 7 (b) f 0 (x) = 5 + 2 x +1 √ x + x5/4 00 (c) f (x) = , f 0 (1) = 5, f (0) = 4 x (d) f 00 (x) = x − 4 − ex , f (0) = 3, f (1) = 2 (a) f 0 (x) = x4 − 7. A particle in motion has acceleration given by the function a(t) = sin t + t3 + 1. Find the position function given that v(0) = 0 and s(0) = 3. 8. An object in motion has acceleration given by a(t) =< 2, et + 4 >. Find the position vector function given that v(0) =< 3, 5 > and r(0) =< 0, 0 >. 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) With what velocity does the object hit the ground? 10. A car traveling at a speed of 30 mph (44 m/s) applies a constant acceleration of 8 m/s2 . How far will the car travel before reaching a speed of 60 mph (88 m/s)? 11. A projectile is fired with a speed of 400 m/s at an angle of elevation of 30◦ . How far away will the projectile land? 12. Calculate the following sums. (a) 5 X (2i2 + 3) i=1 (b) 35 X 3i − 3i+1 i=5 1 c Math 151 WIR, Spring 2010, Benjamin Aurispa (c) 20 X i3 i=3 (d) n X (3i + 2)2 i=1 13. Represent the following sum using sigma notation. 14. Calculate lim n→∞ n X 4 i=1 " 4i 3 n n 2 4i + n # 2 4 9 16 25 400 + + + + ··· + 7 8 9 10 25