Solução_Calculo_Stewart_6e

F.
122 ¤ CHAPTER 3 DIFFERENTIATION RULES
TX.10
35. We are given dθ/dt =2 ◦ /min = π 90
rad/min. By the Law of Cosines,
x 2 =12 2 +15 2 − 2(12)(15) cos θ =369− 360 cos θ
2x dx
dt
=360sinθ
dθ
dt
⇒
dx
dt
=
180 sin θ
x
x = √ 369 − 360 cos 60 ◦ = √ 189 = 3 √ 21,so dx
dt
⇒
dθ
dt .Whenθ =60◦ ,
=
180 sin 60◦
3 √ 21
37. (a) By the Pythagorean Theorem, 4000 2 + y 2 = 2 . Differentiating with respect to t,
we obtain 2y dy
dt
d
dy
=2 . We know that =600ft/s, so when y =3000ft,
dt dt
= √ 4000 2 + 3000 2 = √ 25,000,000 = 5000 ft
and d
dt = y dy
(b) Here tan θ =
dt = 3000
5000
y
4000
⇒
(600) =
1800
5
=360ft/s.
d
dt (tan θ) = d y
dt 4000
⇒
π
90 = π √ √
3 7 π
3 √ 21 = ≈ 0.396 m/min.
21
sec 2 θ dθ
dt = 1 dy
4000 dt
⇒
dθ
dt = cos2 θ dy
4000 dt .When
y =3000ft, dy
4000
=600ft/s, =5000and cos θ = = 4000
dt 5000 = 4 dθ
,so
5 dt = (4/5)2 (600) = 0.096 rad/s.
4000
39. cot θ = x 5
dx
dt = 5π 6
⇒
−csc 2 θ dθ
dt = 1 dx
5 dt
⇒
2 2√3
= 10 π km/min [≈ 130 mi/h]
9
− csc π 2
− π
= 1 dx
3 6 5 dt
⇒
41. We are given that dx
dt
=300km/h. By the Law of Cosines,
y 2 = x 2 +1 2 − 2(1)(x) cos 120 ◦ = x 2 +1− 2x − 1 2
= x 2 + x +1,so
2y dy dx
=2x
dt dt + dx
dt
⇒
dy
dt
=
2x +1
2y
dx
300
.After1 minute, x =
60
dt =5km ⇒
y = √ 5 2 +5+1= √ 31 km ⇒ dy
dt = 2(5) + 1
2 √ 1650
(300) = √ ≈ 296 km/h.
31 31
43. Let the distance between the runner and the friend be .ThenbytheLawofCosines,
2 =200 2 +100 2 − 2 · 200 · 100 · cos θ =50,000 − 40,000 cos θ (). Differentiating
implicitly with respect to t,weobtain2 d
dt
dθ
= −40,000(− sin θ) .NowifD is the
dt
distance run when the angle is θ radians, then by the formula for the length of an arc
on a circle, s = rθ,wehaveD = 100θ,soθ = 1
100 D ⇒ dθ
dt = 1 dD
100 dt = 7 . To substitute into the expression for
100
d
dt , we must know sin θ atthetimewhen =200,whichwefind from (): 2002 =50,000 − 40,000 cos θ
cos θ = 1 ⇒ sin θ = 1 −
1 2
= √ 15
d
. Substituting, we get 2(200)
4 4 4
dt =40,000 √
15 7
⇒
4 100
d/dt = 7 √ 15
4
≈ 6.78 m/s. Whether the distance between them is increasing or decreasing depends on the direction in which
the runner is running.
⇔