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|00000d80| 20 61 6e 64 20 74 68 65 | 20 76 6f 6c 75 6d 65 20 | and the| volume |
|00000d90| 24 56 24 20 6f 66 20 77 | 61 74 65 72 20 69 6e 20 |$V$ of w|ater in |
|00000da0| 74 68 65 20 62 75 63 6b | 65 74 20 24 74 24 20 73 |the buck|et $t$ s|
|00000db0| 65 63 6f 6e 64 73 20 6c | 61 74 65 72 20 69 73 20 |econds l|ater is |
|00000dc0| 67 69 76 65 6e 20 62 79 | 20 5c 5b 20 56 20 3d 20 |given by| \[ V = |
|00000dd0| 31 30 5c 6c 65 66 74 28 | 20 31 20 2d 20 20 5c 66 |10\left(| 1 - \f|
|00000de0| 72 61 63 7b 74 7d 7b 31 | 30 30 7d 20 5c 72 69 67 |rac{t}{1|00} \rig|
|00000df0| 68 74 29 5e 32 20 5c 5d | 20 75 6e 74 69 6c 20 74 |ht)^2 \]| until t|
|00000e00| 68 65 20 62 75 63 6b 65 | 74 20 69 73 20 65 6d 70 |he bucke|t is emp|
|00000e10| 74 79 20 61 74 20 74 69 | 6d 65 20 24 74 20 3d 20 |ty at ti|me $t = |
|00000e20| 31 30 30 24 2e 0a 09 5c | 62 65 67 69 6e 7b 65 6e |100$...\|begin{en|
|00000e30| 75 6d 65 72 61 74 65 7d | 0a 09 5c 69 74 65 6d 20 |umerate}|..\item |
|00000e40| 41 74 20 77 68 61 74 20 | 72 61 74 65 20 69 73 20 |At what |rate is |
|00000e50| 77 61 74 65 72 20 6c 65 | 61 6b 69 6e 67 20 66 72 |water le|aking fr|
|00000e60| 6f 6d 20 74 68 65 20 62 | 75 63 6b 65 74 20 61 66 |om the b|ucket af|
|00000e70| 74 65 72 20 65 78 61 63 | 74 6c 79 20 6f 6e 65 20 |ter exac|tly one |
|00000e80| 6d 69 6e 75 74 65 20 68 | 61 73 20 70 61 73 73 65 |minute h|as passe|
|00000e90| 64 3f 0a 09 5c 69 74 65 | 6d 20 57 68 65 6e 20 69 |d?..\ite|m When i|
|00000ea0| 73 20 74 68 65 20 69 6e | 73 74 61 6e 74 61 6e 65 |s the in|stantane|
|00000eb0| 6f 75 73 20 72 61 74 65 | 20 6f 66 20 63 68 61 6e |ous rate| of chan|
|00000ec0| 67 65 20 6f 66 20 24 56 | 24 20 65 71 75 61 6c 20 |ge of $V|$ equal |
|00000ed0| 74 6f 20 74 68 65 20 61 | 76 65 72 61 67 65 20 63 |to the a|verage c|
|00000ee0| 68 61 6e 67 65 20 6f 66 | 20 24 56 24 20 66 72 6f |hange of| $V$ fro|
|00000ef0| 6d 20 24 74 20 3d 20 30 | 24 20 74 6f 20 24 74 20 |m $t = 0|$ to $t |
|00000f00| 3d 20 31 30 30 24 3f 0a | 09 5c 69 74 65 6d 20 44 |= 100$?.|.\item D|
|00000f10| 6f 65 73 20 74 68 65 20 | 67 69 76 65 6e 20 66 6f |oes the |given fo|
|00000f20| 72 6d 75 6c 61 20 66 6f | 72 20 24 56 24 20 6d 61 |rmula fo|r $V$ ma|
|00000f30| 6b 65 20 73 65 6e 73 65 | 20 77 68 65 6e 20 24 74 |ke sense| when $t|
|00000f40| 20 3d 20 32 30 30 24 3f | 0a 09 5c 65 6e 64 7b 65 | = 200$?|..\end{e|
|00000f50| 6e 75 6d 65 72 61 74 65 | 7d 0a 0a 5c 69 74 65 6d |numerate|}..\item|
|00000f60| 20 28 33 2e 31 2e 33 36 | 29 20 54 68 65 20 66 6f | (3.1.36|) The fo|
|00000f70| 6c 6c 6f 77 69 6e 67 20 | 64 61 74 61 20 64 65 73 |llowing |data des|
|00000f80| 63 72 69 62 65 20 74 68 | 65 20 67 72 6f 77 74 68 |cribe th|e growth|
|00000f90| 20 6f 66 20 74 68 65 20 | 70 6f 70 75 6c 61 74 69 | of the |populati|
|00000fa0| 6f 6e 20 24 50 24 20 28 | 69 6e 20 74 68 6f 75 73 |on $P$ (|in thous|
|00000fb0| 61 6e 64 73 29 20 6f 66 | 20 61 20 63 65 72 74 61 |ands) of| a certa|
|00000fc0| 69 6e 20 63 69 74 79 20 | 64 75 72 69 6e 67 20 74 |in city |during t|
|00000fd0| 68 65 20 31 39 37 30 73 | 2e 20 20 55 73 65 20 74 |he 1970s|. Use t|
|00000fe0| 68 65 20 67 72 61 70 68 | 69 63 61 6c 20 6d 65 74 |he graph|ical met|
|00000ff0| 68 6f 64 20 6f 66 20 45 | 78 61 6d 70 6c 65 20 34 |hod of E|xample 4|
|00001000| 20 74 6f 20 65 73 74 69 | 6d 61 74 65 73 20 69 74 | to esti|mates it|
|00001010| 73 20 72 61 74 65 20 6f | 66 20 67 72 6f 77 74 68 |s rate o|f growth|
|00001020| 20 69 6e 20 31 39 37 35 | 2e 0a 0a 5c 62 65 67 69 | in 1975|...\begi|
|00001030| 6e 7b 74 61 62 75 6c 61 | 72 7d 7b 7c 63 7c 63 7c |n{tabula|r}{|c|c||
|00001040| 63 7c 63 7c 63 7c 63 7c | 63 7c 7d 20 5c 68 6c 69 |c|c|c|c||c|} \hli|
|00001050| 6e 65 0a 59 65 61 72 20 | 26 20 31 39 37 30 20 26 |ne.Year |& 1970 &|
|00001060| 20 31 39 37 32 20 26 20 | 31 39 37 34 20 26 20 31 | 1972 & |1974 & 1|
|00001070| 39 37 36 20 26 20 31 39 | 37 38 20 26 20 31 39 38 |976 & 19|78 & 198|
|00001080| 30 20 5c 5c 20 5c 68 6c | 69 6e 65 0a 24 50 24 20 |0 \\ \hl|ine.$P$ |
|00001090| 26 20 32 36 35 20 26 20 | 32 39 33 20 26 20 33 32 |& 265 & |293 & 32|
|000010a0| 34 20 26 20 33 35 38 20 | 26 20 33 39 35 20 26 20 |4 & 358 |& 395 & |
|000010b0| 34 33 37 20 20 5c 5c 20 | 5c 68 6c 69 6e 65 0a 5c |437 \\ |\hline.\|
|000010c0| 65 6e 64 7b 74 61 62 75 | 6c 61 72 7d 0a 0a 09 0a |end{tabu|lar}....|
|000010d0| 5c 65 6e 64 7b 65 6e 75 | 6d 65 72 61 74 65 7d 0a |\end{enu|merate}.|
|000010e0| 5c 65 6e 64 7b 64 6f 63 | 75 6d 65 6e 74 7d 0a |\end{doc|ument}. |
+--------+-------------------------+-------------------------+--------+--------+
