geomef NOTE Optimization problems often involve are shown in Fig. 10 and on the
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geomef NOTE Optimization problems often involve are shown in Fig. 10 and on the inside back cover of the book. Perimeter 2r+y Volume ae Figure 10 Solutions can be found following the Check Your Understanding 2.5 I. Volume A canvas wind shelter for the beach has a back, two square sides, and a top (Fig. 11). If 96 square feet of canvas is to be used, find the dimensions of the shelter for which the space inside the shelter (the volume) is maximized 2. In Check Your Understanding 1, what are the objective oqua tion and the constraint equation? Figure 11 Wind shelter. EXERCISES 2.5 surface area of the box is minimal. (The surface anat sum of the areas of the five faces of the box.) 1. For what x does the function gx) 10+40x-x have its maximum value? (b) Express the quantity to be minimized as a function (c) Find the optimal values of x and h. 2. Find the maximum value of the function fx)- 12x -x and give the value of x where this maximum occurs give the value of r where this minimum occurs mum value? 3. Find the minimum value or nr).?-6r2 + 40, ,zo.and 4. For what i does the function A)1-24: have its mini- 5. Optimization with Constraint Find the maximum of Q-xy if 6. Optimization with Constraint Find two positive numbers x and 7. Optimization with Constraint Find the minimum of 8. In Exercise 7, can there be a maximum for Q-xi 9. Minimizing Sum Find the positive values of x and y that Road y that maximize xyifxy-2 Figure 12 13. Volume Postal requirements specify that parcels x+y-6 Justify your answer minimize Sx+yif xy 36, and find this minimum value 10. Maximizing a Product Find the positive values of x, y, and : length plus girth of at most 84 inches Consider the probe of finding the dimensions of the square-ended package of greatest volume that is mailable (a) Draw a square-ended rectangular bos. Label cach that maumize Q-xyz, if x + y-land y+:?2.What is 11. Area There are $320 available to fence in a rectangular gar- g for the other three sides costs the square end with the leter x and label the remu this maximum valuc? dimension of the box with the letter h (b) Express the length plus the girth in terms ot den. The fencing for the side of the garden facing the road Determine the objective and constraint equations costs S6 per tsor ig: 12a)Conidr the problem of finding 14. Perimeter Consider the problem of findnse mtcrs (4) Express the quantity to be maximized as a (e) Find the optimal values of x and h t and the fencin the dimensions of the largest possible garden. (a) Determine the objective and constraint equations (b) Express the quantity to be maximized as a function of x (c) Find the optimal values of x and y the rectangular garden of area 100 sqular e the amount of fencing needed to surround small as possible. 12. Volume Figure 12b) shows an open rectangular box with a (a) Draw a picture of a rectangle and selec square base. Consider the problem of finding the values of xand h for which the volume is 32 cubic feet and the total ters for the dimensions Determin Find the optimal values for the dimensions (e) e the objective and constraintExplanation / Answer
7)
x+y=6
=>y =6-x
Q=x2+y2
=>Q=x2+(6-x)2
=>Q=2x2-12x+36
dQ/dx=4x-12 ,d2Q/dx2=4
for critical point , dQ/dx=4x-12=0
=>4x=12
=>x=3
for x=3,d2Q/dx2=4 >0
so Q has minimum value when x=3
y =6-x ,x=3
=>y =6-3
=>y =3
minimum of Q =32+32
=>minimum of Q =18
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12)
(a)
surface area = x2+4xh , volume =x2h
objective function : s =x2+4xh , constraint :x2h =32
(b)
x2h =32
=>h =32/x2
s= x2+4x(32/x2)
=> s= x2+(128/x)
minimize x2+(128/x)
(c)
s= x2+(128/x)
=>ds/dx= 2x-(128/x2) ,d2s/dx2= 2+(256/x3)
for critical point ,ds/dx= 2x-(128/x2)=0
=>2x-(128/x2)=0
=>x3=64
=>x=4
d2s/dx2= 2+(256/43) =6 >0
so s is minimum for x= 4
h =32/x2,x=4
=>h =32/42
=>h=2
therefore optimum values are x= 4 , h=2
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