Answer:
y - 70, x - 73 should be the answer
Step-by-step explanation:
1.8, Problem 37: A lidless cardboard box is to be made with a volume of 4 m3
. Find the
dimensions of the box that requires the least amount of cardboard.
Solution: If the dimensions of our box are x, y, and z, then we’re seeking to minimize
A(x, y, z) = xy + 2xz + 2yz subject to the constraint that xyz = 4. Our first step is to make
the first function a function of just 2 variables. From xyz = 4, we see z = 4/xy, and if we substitute
this into A(x, y, z), we obtain a new function A(x, y) = xy + 8/y + 8/x. Since we’re optimizing
something, we want to calculate the critical points, which occur when Ax = Ay = 0 or either Ax
or Ay is undefined. If Ax or Ay is undefined, then x = 0 or y = 0, which means xyz = 4 can’t
hold. So, we calculate when Ax = 0 = Ay. Ax = y − 8/x2 = 0 and Ay = x − 8/y2 = 0. From
these, we obtain x
2y = 8 = xy2
. This forces x = y = 2, which forces z = 1. Calculating second
derivatives and applying the second derivative test, we see that (x, y) = (2, 2) is a local minimum
for A(x, y). To show it’s an absolute minimum, first notice that A(x, y) is defined for all choices
of x and y that are positive (if x and y are arbitrarily large, you can still make z REALLY small
so that xyz = 4 still). Therefore, the domain is NOT a closed and bounded region (it’s neither
closed nor bounded), so you can’t apply the Extreme Value Theorem. However, you can salvage
something: observe what happens to A(x, y) as x → 0, as y → 0, as x → ∞, and y → ∞. In each
of these cases, at least one of the variables must go to ∞, meaning that A(x, y) goes to ∞. Thus,
moving away from (2, 2) forces A(x, y) to increase, and so (2, 2) is an absolute minimum for A(x, y).
Answer:
1. 87
A77
B24
C93
D47
E 31
F 48
2. 160
A10
B20
C30
D40
E 50
F 60
3. 68
A851
B295
C302
D574
E 112
F 638
4. 174
A3
B4
C5
D 6
E 7
F 8
5. a
A180-a
B180/2
C 179*a
D 179+a
E 60
F pi
Find the complementary angles of the following.
6. 53
A90
B32
C 25
D 45
E 18
F 37
7. 12
A46
B78
C98
D31
E 52
F 64
8. 73.5
A16.5
B34.2
C29.4
D17.7
E 57.9
F 11.3
9. 23.7
A66.3
B70.1
C42.5
D83.9
E 54.8
F 36.2
10. a
A90/a
Ba-90
Ca/2
D90-a
E 90-2
F a^2
Two of the angles are listed. Find the measure of the third angle in each triangle. Similar to example 6
11. 16,42
A256
B421
C107
D510
E 122
F 329
12. 90, 30
A23
B42
C47
D60
E 71
F 89
13. 43, 118
A12
B34
C19
D57
E 11
F 27
14. 60, 60
A15
B90
C29
D60
E 30
F 45
15. 14, 123
A73
B25
C38
D19
E 43
F 13
16. 68, 86
A45
B56
C26
D19
E 32
F 82
17. 55, 77
A18
B37
C62
D57
E 20
F 48
18. a, (a + 80)
A100 - 2a
B112 - 7a
C182 - 2a
D254 - 5a
E 311 - 3a
F 302 - 1a
19. x, (x + 20)
A189 - 7x
B167 - 1x
C114 - 5x
D160 - 2x
E 124 - 2x
F 142 - 1x
20. m, 2m
A172 - 2m
B187 - 7m
C124 - 4m
D143 - 1m
E 164 - 9m
F 180 - 3m
1. C
2. B
3. E
4. D
5. D
6. F
7. B
8. A
9. A
10. B
11. D
12. A
13. D
14. B
15. A
16. C
17. F
18. A
19. D
20. F
Step-by-step explanation:
