/ NeXT Education Software Sampler 1992 Fall / NeXT Education Software Sampler 1992 Fall.iso / Mathematics / Notebooks / CSOMinesCalculus / Chapter3 / ans2.tex

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|00000000| 5c 64 6f 63 75 6d 65 6e | 74 73 74 79 6c 65 5b 31 |\documen|tstyle[1|
|00000010| 31 70 74 2c 66 6c 65 71 | 6e 2c 65 70 73 66 2c 63 |1pt,fleq|n,epsf,c|
|00000020| 61 6c 63 5d 7b 61 72 74 | 69 63 6c 65 7d 0a 5c 6d |alc]{art|icle}.\m|
|00000030| 61 72 6b 72 69 67 68 74 | 7b 43 68 61 70 74 65 72 |arkright|{Chapter|
|00000040| 20 33 3a 20 41 6e 73 77 | 65 72 73 20 32 7d 0a 5c | 3: Answ|ers 2}.\|
|00000050| 62 65 67 69 6e 7b 64 6f | 63 75 6d 65 6e 74 7d 0a |begin{do|cument}.|
|00000060| 0a 5c 42 66 7b 43 68 61 | 70 74 65 72 20 33 3a 20 |.\Bf{Cha|pter 3: |
|00000070| 41 6e 73 77 65 72 73 20 | 32 20 5c 68 66 69 6c 6c |Answers |2 \hfill|
|00000080| 20 4a 61 63 6b 20 4b 2e | 20 43 6f 68 65 6e 20 5c | Jack K.| Cohen \|
|00000090| 68 66 69 6c 6c 20 43 6f | 6c 6f 72 61 64 6f 20 53 |hfill Co|lorado S|
|000000a0| 63 68 6f 6f 6c 20 6f 66 | 20 4d 69 6e 65 73 7d 0a |chool of| Mines}.|
|000000b0| 0a 5c 62 65 67 69 6e 7b | 65 6e 75 6d 65 72 61 74 |.\begin{|enumerat|
|000000c0| 65 7d 0a 5c 69 74 65 6d | 20 24 41 28 72 29 20 3d |e}.\item| $A(r) =|
|000000d0| 20 5c 70 69 20 72 5e 32 | 24 2c 20 73 6f 20 24 5c | \pi r^2|$, so $\|
|000000e0| 66 72 61 63 7b 64 41 7d | 7b 64 72 7d 20 3d 20 32 |frac{dA}|{dr} = 2|
|000000f0| 20 5c 70 69 20 72 24 2e | 20 20 54 68 69 73 20 73 | \pi r$.|  This s|
|00000100| 61 79 73 20 74 68 61 74 | 20 24 5c 66 72 61 63 7b |ays that| $\frac{|
|00000110| 64 41 7d 7b 64 72 7d 20 | 3d 20 43 24 2c 20 74 68 |dA}{dr} |= C$, th|
|00000120| 65 20 63 69 72 63 75 6d | 66 65 72 65 6e 63 65 20 |e circum|ference |
|00000130| 6f 66 20 74 68 65 20 63 | 69 72 63 6c 65 2d 2d 2d |of the c|ircle---|
|00000140| 69 66 20 79 6f 75 20 73 | 74 6f 70 20 74 6f 20 74 |if you s|top to t|
|00000150| 68 69 6e 6b 2c 20 74 68 | 69 73 20 6d 61 6b 65 73 |hink, th|is makes|
|00000160| 20 67 65 6f 6d 65 74 72 | 69 63 20 73 65 6e 73 65 | geometr|ic sense|
|00000170| 2e 0a 0a 5c 69 74 65 6d | 20 20 45 76 61 6c 75 61 |...\item|  Evalua|
|00000180| 74 65 20 74 68 65 20 70 | 72 65 76 69 6f 75 73 20 |te the p|revious |
|00000190| 72 65 73 75 6c 74 20 66 | 6f 72 20 24 5c 66 72 61 |result f|or $\fra|
|000001a0| 63 7b 64 41 7d 7b 64 72 | 7d 20 5c 61 70 70 72 6f |c{dA}{dr|} \appro|
|000001b0| 78 20 32 31 2e 39 39 24 | 20 6d 20 77 68 65 6e 20 |x 21.99$| m when |
|000001c0| 24 72 20 3d 20 33 2e 35 | 24 20 6d 2e 0a 0a 5c 69 |$r = 3.5|$ m...\i|
|000001d0| 74 65 6d 20 28 33 2e 31 | 2e 33 31 29 20 24 41 20 |tem (3.1|.31) $A |
|000001e0| 3d 20 5c 70 69 20 72 5e | 32 2c 20 43 20 3d 20 32 |= \pi r^|2, C = 2|
|000001f0| 20 5c 70 69 20 72 24 2e | 20 20 45 6c 69 6d 69 6e | \pi r$.|  Elimin|
|00000200| 61 74 65 20 24 72 20 3d | 43 20 2f 20 28 32 20 5c |ate $r =|C / (2 \|
|00000210| 70 69 29 24 20 74 6f 20 | 67 65 74 20 24 41 28 43 |pi)$ to |get $A(C|
|00000220| 29 20 3d 20 43 5e 32 2f | 20 28 34 20 5c 70 69 29 |) = C^2/| (4 \pi)|
|00000230| 24 2e 20 20 53 6f 2c 20 | 24 5c 66 72 61 63 7b 64 |$.  So, |$\frac{d|
|00000240| 41 7d 7b 64 43 7d 20 3d | 20 43 20 2f 20 28 32 20 |A}{dC} =| C / (2 |
|00000250| 5c 70 69 29 24 2e 0a 0a | 5c 69 74 65 6d 20 20 24 |\pi)$...|\item  $|
|00000260| 5c 66 72 61 63 7b 64 41 | 7d 7b 64 43 7d 20 5c 61 |\frac{dA|}{dC} \a|
|00000270| 70 70 72 6f 78 20 20 30 | 2e 35 36 24 20 6d 20 77 |pprox  0|.56$ m w|
|00000280| 68 65 6e 20 24 43 20 3d | 20 33 2e 35 24 20 6d 2e |hen $C =| 3.5$ m.|
|00000290| 0a 0a 5c 69 74 65 6d 20 | 28 33 2e 31 2e 33 32 29 |..\item |(3.1.32)|
|000002a0| 20 54 68 65 20 69 6d 70 | 6c 69 63 61 74 69 6f 6e | The imp|lication|
|000002b0| 20 69 73 20 74 68 61 74 | 20 24 72 24 20 69 6e 63 | is that| $r$ inc|
|000002c0| 72 65 61 73 65 73 20 61 | 74 20 61 20 5c 45 6d 7b |reases a|t a \Em{|
|000002d0| 63 6f 6e 73 74 61 6e 74 | 7d 20 72 61 74 65 2c 20 |constant|} rate, |
|000002e0| 73 6f 20 24 72 20 3d 20 | 35 20 74 24 20 66 74 2e |so $r = |5 t$ ft.|
|000002f0| 20 20 54 68 75 73 20 24 | 41 28 74 29 20 3d 20 5c |  Thus $|A(t) = \|
|00000300| 70 69 20 72 5e 32 20 3d | 20 32 35 20 5c 70 69 20 |pi r^2 =| 25 \pi |
|00000310| 74 5e 32 24 20 66 74 24 | 5e 32 24 20 61 6e 64 20 |t^2$ ft$|^2$ and |
|00000320| 24 5c 66 72 61 63 7b 64 | 41 7d 7b 64 74 7d 20 3d |$\frac{d|A}{dt} =|
|00000330| 20 35 30 20 5c 70 69 20 | 74 24 20 20 66 74 24 5e | 50 \pi |t$  ft$^|
|00000340| 32 24 2f 73 65 63 2e 20 | 20 57 68 65 6e 20 24 74 |2$/sec. | When $t|
|00000350| 20 3d 20 31 30 24 2c 20 | 77 65 20 68 61 76 65 20 | = 10$, |we have |
|00000360| 24 5c 66 72 61 63 7b 64 | 41 7d 7b 64 74 7d 20 3d |$\frac{d|A}{dt} =|
|00000370| 20 35 30 30 20 5c 70 69 | 24 20 20 66 74 24 5e 32 | 500 \pi|$  ft$^2|
|00000380| 24 2f 73 65 63 2e 0a 0a | 5c 69 74 65 6d 20 28 47 |$/sec...|\item (G|
|00000390| 65 6e 65 72 61 6c 69 7a | 61 74 69 6f 6e 20 6f 66 |eneraliz|ation of|
|000003a0| 20 45 78 61 6d 70 6c 65 | 20 36 29 20 0a 09 5c 62 | Example| 6) ..\b|
|000003b0| 65 67 69 6e 7b 65 6e 75 | 6d 65 72 61 74 65 7d 0a |egin{enu|merate}.|
|000003c0| 09 5c 69 74 65 6d 20 41 | 74 20 74 68 65 20 6d 61 |.\item A|t the ma|
|000003d0| 78 69 6d 75 6d 20 68 65 | 69 67 68 74 2c 20 24 76 |ximum he|ight, $v|
|000003e0| 20 3d 20 30 24 2c 20 73 | 6f 20 24 74 20 3d 20 74 | = 0$, s|o $t = t|
|000003f0| 5f 7b 74 6f 70 7d 20 3d | 20 76 5f 30 20 2f 20 67 |_{top} =| v_0 / g|
|00000400| 24 20 61 6e 64 20 24 73 | 5f 7b 74 6f 70 7d 20 3d |$ and $s|_{top} =|
|00000410| 20 76 5f 30 5e 32 2f 20 | 28 20 32 20 67 29 24 2e | v_0^2/ |( 2 g)$.|
|00000420| 0a 09 5c 69 74 65 6d 20 | 41 74 20 67 72 6f 75 6e |..\item |At groun|
|00000430| 64 20 6c 65 76 65 6c 2c | 20 24 73 20 3d 20 30 24 |d level,| $s = 0$|
|00000440| 2c 20 73 6f 20 24 30 20 | 3d 20 2d 67 74 5e 32 2f |, so $0 |= -gt^2/|
|00000450| 32 20 2b 20 76 5f 30 20 | 74 20 3d 20 74 28 76 5f |2 + v_0 |t = t(v_|
|00000460| 30 20 2d 20 67 20 74 2f | 32 29 24 2e 20 20 54 68 |0 - g t/|2)$.  Th|
|00000470| 65 20 24 74 20 3d 20 30 | 24 20 72 6f 6f 74 20 63 |e $t = 0|$ root c|
|00000480| 6f 72 72 65 73 70 6f 6e | 64 73 20 74 6f 20 74 68 |orrespon|ds to th|
|00000490| 65 20 69 6e 69 74 69 61 | 6c 20 74 69 6d 65 2c 20 |e initia|l time, |
|000004a0| 73 6f 20 74 68 65 20 6f | 74 68 65 72 20 72 6f 6f |so the o|ther roo|
|000004b0| 74 20 67 69 76 65 73 20 | 74 68 65 20 66 69 6e 61 |t gives |the fina|
|000004c0| 6c 20 74 69 6d 65 2c 20 | 24 74 5f 66 20 3d 20 32 |l time, |$t_f = 2|
|000004d0| 76 5f 30 2f 67 24 2e 20 | 20 54 68 75 73 2c 20 74 |v_0/g$. | Thus, t|
|000004e0| 68 65 20 66 69 6e 61 6c | 20 76 65 6c 6f 63 69 74 |he final| velocit|
|000004f0| 79 20 69 73 20 24 76 5f | 66 20 3d 20 2d 20 67 20 |y is $v_|f = - g |
|00000500| 5c 63 64 6f 74 20 32 76 | 5f 30 2f 67 20 2b 20 76 |\cdot 2v|_0/g + v|
|00000510| 5f 30 20 3d 20 2d 20 76 | 5f 30 24 2e 0a 09 5c 69 |_0 = - v|_0$...\i|
|00000520| 74 65 6d 20 20 54 68 65 | 20 69 6e 69 74 69 61 6c |tem  The| initial|
|00000530| 20 61 6e 64 20 66 69 6e | 61 6c 20 76 65 6c 6f 63 | and fin|al veloc|
|00000540| 69 74 69 65 73 20 61 72 | 65 20 65 71 75 61 6c 20 |ities ar|e equal |
|00000550| 69 6e 20 6d 61 67 6e 69 | 74 75 64 65 20 61 6e 64 |in magni|tude and|
|00000560| 20 6f 70 70 6f 73 69 74 | 65 20 69 6e 20 64 69 72 | opposit|e in dir|
|00000570| 65 63 74 69 6f 6e 20 28 | 6e 6f 20 6d 61 74 74 65 |ection (|no matte|
|00000580| 72 20 77 68 61 74 20 74 | 68 65 20 69 6e 69 74 69 |r what t|he initi|
|00000590| 61 6c 20 76 65 6c 6f 63 | 69 74 79 20 69 73 2d 2d |al veloc|ity is--|
|000005a0| 2d 74 68 65 72 65 20 69 | 73 20 6e 6f 74 68 69 6e |-there i|s nothin|
|000005b0| 67 20 73 70 65 63 69 61 | 6c 20 61 62 6f 75 74 20 |g specia|l about |
|000005c0| 74 68 65 20 74 65 78 74 | 27 73 20 63 68 6f 69 63 |the text|'s choic|
|000005d0| 65 20 6f 66 20 39 36 20 | 66 74 2f 73 65 63 29 2e |e of 96 |ft/sec).|
|000005e0| 0a 09 5c 69 74 65 6d 20 | 54 68 65 20 64 69 6d 65 |..\item |The dime|
|000005f0| 6e 73 69 6f 6e 73 20 6f | 66 20 74 68 65 20 72 65 |nsions o|f the re|
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|00000610| 3d 20 20 76 5f 30 5e 32 | 2f 28 32 20 67 29 24 20 |=  v_0^2|/(2 g)$ |
|00000620| 61 72 65 20 24 28 5c 66 | 72 61 63 7b 4c 7d 7b 54 |are $(\f|rac{L}{T|
|00000630| 7d 29 5e 32 20 5c 64 69 | 76 20 5c 66 72 61 63 7b |})^2 \di|v \frac{|
|00000640| 4c 7d 7b 54 5e 32 7d 20 | 3d 20 4c 24 20 77 68 69 |L}{T^2} |= L$ whi|
|00000650| 63 68 20 69 73 20 63 6f | 72 72 65 63 74 2e 20 20 |ch is co|rrect.  |
|00000660| 20 54 68 65 20 64 69 6d | 65 6e 73 69 6f 6e 73 20 | The dim|ensions |
|00000670| 6f 66 20 24 76 5f 66 20 | 3d 20 2d 20 76 5f 30 24 |of $v_f |= - v_0$|
|00000680| 20 61 72 65 20 6f 62 76 | 69 6f 75 73 6c 79 20 74 | are obv|iously t|
|00000690| 68 65 20 64 69 6d 65 6e | 73 69 6f 6e 73 20 6f 66 |he dimen|sions of|
|000006a0| 20 61 20 76 65 6c 6f 63 | 69 74 79 2c 20 73 6f 20 | a veloc|ity, so |
|000006b0| 74 68 69 73 20 63 68 65 | 63 6b 73 20 61 73 20 77 |this che|cks as w|
|000006c0| 65 6c 6c 2e 09 5c 65 6e | 64 7b 65 6e 75 6d 65 72 |ell..\en|d{enumer|
|000006d0| 61 74 65 7d 0a 0a 5c 69 | 74 65 6d 20 20 52 65 20 |ate}..\i|tem  Re |
|000006e0| 74 68 65 20 69 6e 76 65 | 72 73 65 20 73 71 75 61 |the inve|rse squa|
|000006f0| 72 65 20 6c 61 77 3a 0a | 09 5c 62 65 67 69 6e 7b |re law:.|.\begin{|
|00000700| 65 6e 75 6d 65 72 61 74 | 65 7d 0a 09 5c 69 74 65 |enumerat|e}..\ite|
|00000710| 6d 20 59 65 73 21 20 20 | 57 68 65 6e 20 24 73 20 |m Yes!  |When $s |
|00000720| 3d 20 30 24 2c 20 77 65 | 20 73 65 65 20 66 72 6f |= 0$, we| see fro|
|00000730| 6d 20 74 68 65 20 67 69 | 76 65 6e 20 66 6f 72 6d |m the gi|ven form|
|00000740| 75 6c 61 20 66 6f 72 20 | 24 73 24 20 74 68 61 74 |ula for |$s$ that|
|00000750| 20 24 76 5f 66 5e 32 20 | 3d 20 76 5f 30 5e 32 24 | $v_f^2 |= v_0^2$|
|00000760| 2c 20 73 6f 20 61 67 61 | 69 6e 20 74 68 65 20 66 |, so aga|in the f|
|00000770| 69 6e 61 6c 20 61 6e 64 | 20 69 6e 69 74 69 61 6c |inal and| initial|
|00000780| 20 76 65 6c 6f 63 69 74 | 69 65 73 20 68 61 76 65 | velocit|ies have|
|00000790| 20 74 68 65 20 73 61 6d | 65 20 6d 61 67 6e 69 74 | the sam|e magnit|
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|000007b0| 68 65 20 6d 61 78 69 6d | 75 6d 20 68 65 69 67 68 |he maxim|um heigh|
|000007c0| 74 2c 20 24 76 20 3d 20 | 30 24 2c 20 73 6f 20 24 |t, $v = |0$, so $|
|000007d0| 73 5f 7b 74 6f 70 7d 20 | 3d 20 5c 66 72 61 63 7b |s_{top} |= \frac{|
|000007e0| 52 20 76 5f 30 5e 32 7d | 7b 32 20 67 20 52 20 2d |R v_0^2}|{2 g R -|
|000007f0| 20 76 5f 30 5e 32 7d 24 | 2e 0a 09 0a 09 5c 69 74 | v_0^2}$|.....\it|
|00000800| 65 6d 20 57 69 74 68 20 | 74 68 65 20 67 69 76 65 |em With |the give|
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|00000820| 6f 6e 73 74 61 6e 74 20 | 66 6f 72 63 65 20 6c 61 |onstant |force la|
|00000830| 77 20 67 69 76 65 73 20 | 24 73 5f 7b 74 6f 70 7d |w gives |$s_{top}|
|00000840| 20 3d 20 31 34 34 24 20 | 66 74 20 61 6e 64 20 74 | = 144$ |ft and t|
|00000850| 68 65 20 69 6e 76 65 72 | 73 65 20 73 71 75 61 72 |he inver|se squar|
|00000860| 65 20 72 65 73 75 6c 74 | 20 67 69 76 65 73 20 31 |e result| gives 1|
|00000870| 34 34 2e 30 30 31 20 66 | 74 2e 20 20 54 68 75 73 |44.001 f|t.  Thus|
|00000880| 20 66 6f 72 20 76 65 6c | 6f 63 69 74 69 65 73 20 | for vel|ocities |
|00000890| 6f 66 20 74 68 69 73 20 | 6f 72 64 65 72 20 6f 66 |of this |order of|
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|000008c0| 66 65 63 74 20 6f 66 20 | 74 68 65 20 69 6e 76 65 |fect of |the inve|
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|000008f0| 68 69 67 68 20 70 72 65 | 63 69 73 69 6f 6e 20 69 |high pre|cision i|
|00000900| 73 20 64 65 6d 61 6e 64 | 65 64 2e 0a 09 5c 65 6e |s demand|ed...\en|
|00000910| 64 7b 65 6e 75 6d 65 72 | 61 74 65 7d 0a 0a 5c 69 |d{enumer|ate}..\i|
|00000920| 74 65 6d 20 52 65 20 74 | 68 65 20 77 61 74 65 72 |tem Re t|he water|
|00000930| 20 62 75 63 6b 65 74 20 | 70 72 6f 62 6c 65 6d 3a | bucket |problem:|
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|00000960| 67 69 76 65 6e 20 66 75 | 6e 63 74 69 6f 6e 20 65 |given fu|nction e|
|00000970| 78 70 61 6e 64 73 20 74 | 6f 20 74 68 65 20 71 75 |xpands t|o the qu|
|00000980| 61 64 72 61 74 69 63 20 | 24 31 30 20 2d 20 74 2f |adratic |$10 - t/|
|00000990| 35 20 2b 20 74 5e 32 2f | 31 30 30 30 24 2c 20 73 |5 + t^2/|1000$, s|
|000009a0| 6f 20 24 56 27 28 74 29 | 20 3d 20 2d 31 2f 35 20 |o $V'(t)| = -1/5 |
|000009b0| 2b 20 74 2f 35 30 30 24 | 2e 20 20 53 69 6e 63 65 |+ t/500$|.  Since|
|000009c0| 20 24 74 24 20 69 73 20 | 6d 65 61 73 75 72 65 64 | $t$ is |measured|
|000009d0| 20 69 6e 20 73 65 63 6f | 6e 64 73 2c 20 77 65 20 | in seco|nds, we |
|000009e0| 65 76 61 6c 75 61 74 65 | 20 24 56 28 36 30 29 20 |evaluate| $V(60) |
|000009f0| 3d 20 38 2f 35 20 3d 20 | 31 2e 36 24 2e 0a 09 5c |= 8/5 = |1.6$...\|
|00000a00| 69 74 65 6d 20 54 68 65 | 20 61 76 65 72 61 67 65 |item The| average|
|00000a10| 20 63 68 61 6e 67 65 20 | 69 73 20 24 28 56 28 31 | change |is $(V(1|
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|00000a30| 30 30 20 2d 20 30 29 20 | 3d 28 30 20 2d 20 31 30 |00 - 0) |=(0 - 10|
|00000a40| 29 2f 31 30 30 20 3d 20 | 2d 20 31 2f 31 30 24 2e |)/100 = |- 1/10$.|
|00000a50| 20 20 45 71 75 61 74 69 | 6e 67 20 74 68 69 73 20 |  Equati|ng this |
|00000a60| 74 6f 20 24 56 27 28 74 | 29 24 20 67 69 76 65 73 |to $V'(t|)$ gives|
|00000a70| 20 74 68 65 20 6c 69 6e | 65 61 72 20 65 71 75 61 | the lin|ear equa|
|00000a80| 74 69 6f 6e 20 24 2d 31 | 2f 31 30 20 3d 20 2d 31 |tion $-1|/10 = -1|
|00000a90| 2f 35 20 2b 20 74 2f 35 | 30 30 24 20 77 69 74 68 |/5 + t/5|00$ with|
|00000aa0| 20 73 6f 6c 75 74 69 6f | 6e 20 24 74 20 3d 20 35 | solutio|n $t = 5|
|00000ab0| 30 24 20 73 65 63 2e 0a | 09 5c 69 74 65 6d 20 4e |0$ sec..|.\item N|
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|00000b80| 20 49 6e 20 74 68 69 73 | 20 63 61 73 65 2c 20 66 | In this| case, f|
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|00000ba0| 56 28 30 29 20 3d 20 31 | 30 24 20 61 73 20 61 73 |V(0) = 1|0$ as as|
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|00000d10| 54 74 7b 31 37 2e 30 30 | 33 31 7d 20 77 68 69 63 |Tt{17.00|31} whic|
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|00000df0| 7d 0a                   |                         |}.      |        |
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