·111 12:16AM Select text 1. Find two positive numbers such that their product is

Question

·111 12:16AM Select text 1. Find two positive numbers such that their product is 192 and the sum of the first plus three times the second is a minimum. 2. Find two positive numbers such that the sum of the first and twice the second is 100 and their product is a maximum. 3. A gardener wants to make a rectangular enclosure using a wall as one side and 120 m of fencing for the other three sides. Express the area in terms of x and find the value of x that gives the greatest area. 4. A rectangle has a perimeter of 80 cm. If its width is x, express its length and area in terms of x. and find the maximum area. 5. Suppose you had 102 m of fencing to make two side-by-side enclosures as shown. What is the maximum area that you could enclose? 6. Suppose you had to use exactly 200 m of fencing to make either one square enclosure or two separate square enclosures of any size you wished. What plan would give you the least area? What plan would give you the greatest area? 7. A piece of wire 40 cm long is to be cut into two pieces. One piece will be bent to form a (a) Find the lengths of the two pieces that cause the sum of the area of the circle and the area (b) How could you make the total area of the circle and the square a maximum? 8. Four feet of wire is to be used to form a square and a circle. How much of the wire should circle; the other will be bent to form a square. of the square to be a minimum. be used for the square and how much should be used for the circle to enclose the maximum total area? 9. The combined perimeter of an equilateral triangle and a square is 10. Find the dimensions of the triangle and square that produce a minimum total area. 10. The combined perimeter of a circle and a square is 16. Find the dimensions of the circle . A manufacturer wants to design an open box having a square base and a surface area 12. A rectangular page is to contain 24 sq. in. of print. The margins at the top and bottom of and square that produce a minimum total area. of 108 square inches. What dimensions will produce a box with maximum volume? the page are each inches. The margins on each side are 1 inch. What should the dimensions of the page be so that the least amount of paper is used? 13. A tank with a rectangular base and rectangular sides is open at the top. It is to be constructed so that its width is 4 meters and its volume is 36 cubic meters. If building the tank costs S10/sq. m. for the base and $5/sq. m. for the sides, what is the cost of the least expensive tank, and whata

Explanation / Answer

1. Let the numbers are x and y

x y=192

x=192/y

sum= x+3y

S(y)=(192/y)+3y

S'(y)=(-192/y 2)+3

S''(y)=384y 3>0, so the sum is minimum

S'(y)=0

(-192/y 2)+3=0

y 2=64

y=8

xy=192

x=192/8=24

numbers are 24 and 8

2. Let the numbers are x and y

x+2y=100

x=100-2y

Product=x y= (100-2y)y= 100y-2y 2

P'=100-4y

P''=-4<0, so product is maximum

P'=0

100-4y=0

y=25

x=100-2(25)=50

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