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Chapter 3 74

# Chapter 3 74 - 224 CHAPTER 3 DIFFERENTIATION RULES dV 19 2...

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Unformatted text preview: 224 CHAPTER 3 DIFFERENTIATION RULES dV 19. 2 If C = the rate at which water is pumped in, then E : C — 10,000, where I V = émﬂh is the volume at time t. By similar triangles, g = g— :> 6 _ 1 _ 1 1 2 7r dV 7r . dh i 7" 5h => V—57l'(§h) hZEhs 2 EZEhZ‘d—t. Wh h — 200 dh — 20 ' 7’ 2 en — cm. a — cm/mln, so C — 10,000 = 5(200) (20) : C : 10,000 + W7: :3 289,253 cm3/min. 20 3 . . . 3 b . T By s1milar triangles, 1 : hi so b : 3h. The trough has volume 1 V : §bh(10) : 5(3h)h :15};2 => 12 = Z—V : [email protected] i t dt dh 2 dh 2 4 —=—.Wh :l.__: 2— ‘, dt 5h enh 2 dt 55 5mm“ 21. The ﬁgure is labeled in meters. The area A of a trapezoid is %(base1 + baseg)(height), and the volume V of the 10—meter—long trough is 10A. Thus, the volume of the trapezoid with height h is V : (10); [0.3 + (0.3 + 2a)] h, By similar triangles, % 2 903; : %, so dV dV dh dh : : . : h2.N ——=—— => 0223 100— => 2a h :> V 5(06+h)h 3h+5 0w dt dh dt (+ )dt dh 0.2 dh 0.2 0.2 , 1 . 10 . E _ 3 + 10h' Whenh — 0.3. E — 3+ 10(03) — —6— m/mln — 30 m/mln or 3 cm/min. 22. _I 34 ’I The ﬁgure is drawn without the top 3 feet. 2 _ V 1b 12h20 10(b+12)h (if "1 t' l 2 — : an , rom s1m1 ar rian es, 1 V-V 2< + H > g |<—6——>I<——12—>I<——-16 ——>1 0: 6 y 16 8 8h 11h _:_ _:—:— : :h 12 —=12 ——.Th, h andh 6 3,sob \$+12+y + +3 + 3 us 11h 110h2 dV 220 dh : —— : .:—-: 4 —— ——.Wh h=5, V 10(24+ 3 >h 240h-I- 3 andsoOS dt <2 0+ 3 h) dt en dh 0.8 3 . -— : : —— % 0.00132 ft . dt 240 + 5(220/3) 2275 mm 2 3 , dv_3._12_;ﬂ_lh. 23. Wearegwenthatﬁ —30ft /m1n.V— 37W“ h! 37r(2 h— 12 dV dV dh 7rh2 dh dh 120 __ : ___ = ___. — = —. Wh Z>dt dhdtz>3 4dt=>dt «h? en 120 ...
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