Unformatted text preview: SECTION 4.7 OPTIMIZATION PROBLEMS 365 d5 1 (3056 1 . . .
——  , 0 — —>  3 . — 0 —> c050 = —. The First Derivative
(b) d6 —0 when cscO— x/gcot 0 sinB \/— smi9 J5
Test shows that the minimum surface area occurs when 0 : c0s’1(%) z 55°.
(c) If cos 0 = i. then cot t9 : «ii and csc 0 = % so the surface area is
,, . 3.21 2&3; ___3_2 i2
‘/3 \/2 Szﬁsh—55ﬁ+3$ 2%—63h 2\/§Ls+2\/§s
:6sh+ nfs 21:63(h+ 2—fs)
l
40. 15 km/h N Let t be the time, in hours. after 2:00 RM. The position of the boat heading
W E south at time t is (07 720t). The position of the boat heading east at time t
is (A15 + 1515. 0). If D(t) is the distance between the boats at time t, we
minimize f(t) : [pm]2 : 2022:2 + 15% —1)2.
20 Mb I E f’(t) = 800t + 450(t , 1) _ 1250i: — 450: 0 when t: {12—5050 , _.0 36 h.
S 0.36 h x goh'ﬂ = 21.6 min = 21 min 36 5. Since f"(t) > 0. this gives a
minimum, so the boats are closest together at 2:21 :36 RM.
/ 2 2 7 1
41. HereT(m):$—+5+5 (m)— ‘T _0 9 8m_6i/x2+25 6 8 61/302 + 25 8 x 161’2 — 9(272 I 25) 15 a: 3/51. But 3/5— > 5 s0 T has no critical number. Since T(O) % 1.46 and T(5) % 118. he should row directly to B. 42. B In isosceles triangle AOB. LO 2 1800 i 0 i 0. so ZBOC : 20. The
‘ distance rowed is 4 cos 0 while the distance walked is the length of arc
BC : 2(26) : 40. The time taken is given by
A C
4
T(0) = C356 +416 :2c056+0. 0363 g. T’(0):i2sin0+1=0 <=> sinO: => 92’ l 1
2 6 ' Check the value ofT at 0 : g and at the endpoints of the domain of T; that is, 9 : 0 and 0 = %. T(O) : 2,
T(%) : \/§ +% m 2.26, and T(%) : g m 1.57. Therefore, the minimum value ofT is g when 0 : g; that is, the woman should walk all the way. Note that T"(6) = —2 cos 9 < 0 for 0 S 9 < 325. so 19 : % gives a maximum time. 43. The total illumination is 1(ac) = % + —k— 0 < a: < 10. Then 3k m2 (10 — ac)2 k
—6l~c 2k:
{g—Xﬂmoiiﬁg I’(m):?+m:0 :> 6k(10—x)3:2km3 :> ‘0 3(10—m)3:x3 => \3/§(107$):x 2 ioV—S/Emmc
=>10\3/?_)=$+\3/§$ => 10%:(1+\P/§)m => 109/3
1+??? O<ac<10. x: R: 5.9 ft. This gives a minimum since 1"(x) > 0 for ...
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 Spring '10
 Ban
 Calculus

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