Optimization Multiple Choice Problems for Practice!
Since optimization is essentially an application for differentiation, some of these multiple choice questions will be differentiation questions. Determining the maximums and minimums of a function is the main step in finding the "optimal" solution.
1) Find the absolute maximum and minimum values of the function: f(x)=x^3 - 3x + 1 -1/2</= x </=4
(a) y=17 & 3 (b) y=17 & 0 (c) y= 17 & -3 (d) y= 1 & -3
2) We need to enclose a field with a fence. We have 500 feet of fencing material and a building is on one side of the field and so won’t need any fencing. Determine the dimensions of the field that will enclose the largest area.
(a) x=250 y=125 (b) x=150 y=200 (c) x=125 y=100 (d) x=200 y= 150
3) We want to construct a box whose base length is 3 times the base width. The material used to build the top and bottom cost $10/ft2 and the material used to build the sides cost $6/ft2. If the box must have a volume of 50ft3 determine the dimensions that will minimize the cost to build the box.
(a) w=1.8821, h=4.7050, l=5.6463 (b)w=1.2643, h=5.2637, l= 6.2431
(c) w=5.2656, h=5.3452, l=6.6478 (d)w=6.4543, h=7.4353, l=3.2758
4) for the scenario above, find the minimum cost
(a)$701.55 (b) $637.60 (c) $561.25 (d) $882.89
5)Find the point on the parabola y^2=2x that is closest to the point (1,4)
(a) (1,4) (b) (2,4) (c) (4,1) (d) (2,2)
6)Find the largest area of the largest rectangle that can be inscribed in a semicircle of radius r.
(a) 3r^2 (b) r^2 (c)2r^2 (d) .5r^2
7) You decide to walk from point A (see figure below) to point C. To the south of the road through BC, the terrain is difficult and you can only walk at 3 km/hr. However, along the road BC you can walk at 5 km/hr. The distance from point A to the road is 5 km. The distance from B to C is 10 km. What path you have to follow in order to arrive at point C in the shortest ( minimum ) time possible?
1) Find the absolute maximum and minimum values of the function: f(x)=x^3 - 3x + 1 -1/2</= x </=4
(a) y=17 & 3 (b) y=17 & 0 (c) y= 17 & -3 (d) y= 1 & -3
2) We need to enclose a field with a fence. We have 500 feet of fencing material and a building is on one side of the field and so won’t need any fencing. Determine the dimensions of the field that will enclose the largest area.
(a) x=250 y=125 (b) x=150 y=200 (c) x=125 y=100 (d) x=200 y= 150
3) We want to construct a box whose base length is 3 times the base width. The material used to build the top and bottom cost $10/ft2 and the material used to build the sides cost $6/ft2. If the box must have a volume of 50ft3 determine the dimensions that will minimize the cost to build the box.
(a) w=1.8821, h=4.7050, l=5.6463 (b)w=1.2643, h=5.2637, l= 6.2431
(c) w=5.2656, h=5.3452, l=6.6478 (d)w=6.4543, h=7.4353, l=3.2758
4) for the scenario above, find the minimum cost
(a)$701.55 (b) $637.60 (c) $561.25 (d) $882.89
5)Find the point on the parabola y^2=2x that is closest to the point (1,4)
(a) (1,4) (b) (2,4) (c) (4,1) (d) (2,2)
6)Find the largest area of the largest rectangle that can be inscribed in a semicircle of radius r.
(a) 3r^2 (b) r^2 (c)2r^2 (d) .5r^2
7) You decide to walk from point A (see figure below) to point C. To the south of the road through BC, the terrain is difficult and you can only walk at 3 km/hr. However, along the road BC you can walk at 5 km/hr. The distance from point A to the road is 5 km. The distance from B to C is 10 km. What path you have to follow in order to arrive at point C in the shortest ( minimum ) time possible?
NOTICE***Top right corner letter should be point B not point A, sorry for typo
(a) From A to C (b) Halfway up to B then directly to C (c) 3.75km up road AB then directly to C (d) from A to point P such BP = 3.75 km then procced along the road to C directly
8)A window is being built and the bottom is a rectangle and the top is a semicircle. If there is 12 meters of framing materials what must the dimensions of the window be to let in the most light?
8)A window is being built and the bottom is a rectangle and the top is a semicircle. If there is 12 meters of framing materials what must the dimensions of the window be to let in the most light?
(a)h=1.6803 r=3.3606 (b) h=3.3606 r=1.6803 (c) h= 4.8608 r= 2.4304 (d) h=2.4304, r=4.8608
9)A 2 feet piece of wire is cut into two pieces and once piece is bent into a square and the other is bent into an equilateral triangle. Where should the wire be cut so that the total area enclosed by both shapes is at the minimum?
(a) x=0.8699ft (b) x= 0.8456 (c) 0.8269 (d) 0.8112
10) From number 9 above, where should the wire cut so that the total enclosed area by both shapes is at the maximum?
(a) all wire used for triangle (b) half of wire used for triangle and half for square (c) all wire used for square (d) i have no clue
9)A 2 feet piece of wire is cut into two pieces and once piece is bent into a square and the other is bent into an equilateral triangle. Where should the wire be cut so that the total area enclosed by both shapes is at the minimum?
(a) x=0.8699ft (b) x= 0.8456 (c) 0.8269 (d) 0.8112
10) From number 9 above, where should the wire cut so that the total enclosed area by both shapes is at the maximum?
(a) all wire used for triangle (b) half of wire used for triangle and half for square (c) all wire used for square (d) i have no clue
