Applied Max and Min (Optimization) 1. If you have 100 feet of fencing and you want to enclose a rectangular area up against a long, straight wall, what is the largest area you can enclose? Title: Oct 9 8:27 AM (1 of 30)
2. A rectangular area of 3200 ft 2 is to be fenced off. Two opposite sides will use fencing costing $2 per foot and the remaining sides will use fencing costing $1 per foot. Find the dimensions of the rectangle of least cost. Title: Oct 9 8:27 AM (2 of 30)
3. A square bottomed box with a top has a fixed volume of 500cm3. What dimensions minimize the surface area? Title: Oct 9 8:28 AM (3 of 30)
4. A square bottomed box without a top has a fixed volume of 500cm3. What dimensions minimize the surface area? Title: Oct 9 8:29 AM (4 of 30)
5. Which point on the parabola y = 2 xis closest to the point (1, 0)? Title: Oct 9 8:30 AM (5 of 30)
Title: Oct 20 3:36 PM (6 of 30)
6. Which point on the curve y = 4 2 xis closest to the point (1, 2)? Title: Oct 9 8:30 AM (7 of 30)
Title: Oct 11 9:10 AM (8 of 30)
Title: Oct 11 8:59 AM (9 of 30)
7. A closed cylindrical can is to have surface area 210cm 2. Find the height of the can with maximum volume. Title: Oct 9 8:31 AM (10 of 30)
Title: Oct 11 8:41 AM (11 of 30)
8. A cylindrical can, open at the top, is to have volume 500cm3. Find the height and radius that minimize the amount of material needed to manufacture the can. Title: Oct 9 8:31 AM (12 of 30)
Title: Oct 12 8:26 AM (13 of 30)
9. On the same side of a straight river are two towns, and the townspeople want to build a pumping station, S, that supplies water to them. The pumping station is to be at the river s edge with pipes extending straight to the two towns. Where should the pumping station be located to minimize the total length of pipe? Title: Oct 9 8:32 AM (14 of 30)
Title: Oct 12 9:36 AM (15 of 30)
10. Your snowmobile is out of gas and it is 9:30 p.m. You are stranded in a snowy field 3 miles due south of a major highway which runs in the east west direction. The nearest gas station on the highway is 6 miles east of your position and it closes at midnight. You can walk at a rate of 4 miles per hour on the road, but only 2 miles per hour through the snow. Can you make it to the gas station before it closes? What route is the fastest? Title: Oct 12 8:18 AM (16 of 30)
Title: Oct 13 8:29 AM (17 of 30)
Title: Oct 13 2:44 PM (18 of 30)
Title: Oct 13 8:41 AM (19 of 30)
11. A rectangle is bounded by the x and y axes and the line What length and width should the rectangle have so that its area is a maximum? Title: Oct 12 8:18 AM (20 of 30)
12. A right triangle is formed in the first quadrant by the x and y axes and a line through the point (1,2). Find the vertices of the triangle so that its area is a minimum. Title: Oct 12 8:18 AM (21 of 30)
Title: Oct 19 8:47 AM (22 of 30)
Title: Oct 12 3:27 PM (23 of 30)
13. Find the dimensions of the rectangle of largest area that has its base on the x axis and its other two vertices above the x axis and lying on the parabola Title: Oct 12 8:18 AM (24 of 30)
Title: Oct 19 2:31 PM (25 of 30)
14. A rectangle is drawn with sides parallel to the coordinate axes and with its upper two vertices on the parabola and its lower two vertices on the parabola What is the maximum possible area of the rectangle? Title: Oct 12 8:19 AM (26 of 30)
15. A drainage channel is to be made so that its cross section is a trapezoid with equally sloping sides. If the sides and bottom all have length of 5 feet, how should the angle be chosen to yield the greatest cross sectional area? Title: Oct 12 8:19 AM (27 of 30)
X 5 5 h 5 Area of a trapezoid = h( a + b) 2 where a and b are the parallel sides and h is the perpendicular distance between them We can use right triangle trig to find h and X in terms of θ. h X 5 h 5 = sinθ h = 5sinθ X 5 = cos θ x = 5cosθ So the Area of a trapezoid = h( a + b) 2 = 5sinθ( 5 + 5 + 2*5cosθ) 2 A = 25sinθ + 25sinθcosθ use product rule to find the derivative of this term A'(x ) is ugly!!! A'(x) = 25sin2θ + 25cos 2θ+ 25cosθ Use a calculator to solve for the zeros and do the sign line. Title: Oct 13 3:45 PM (28 of 30)
16. Where on the curve does the tangent line have greatest slope? ANSWER: The slope of the curve is a maximum at the point ( 1, 0.5) For hints see the next page. Title: Oct 12 8:19 AM (29 of 30)
1. We need to find the maximum value of the slope of the function. 2. The slope of any function is the derivative of the function. If, 3. If we want to maximize the slope, we need to maximize the function Let's call this function S for slope so we don't get too confused!!!! 4. We all know how to maximize a function, so go ahead and maximize Title: Oct 13 4:24 PM (30 of 30)