Example7 A spot light is on the ground 20 ft away from a wall and a 6 ft tall person is walking towardsthe wallat a rate of 2.5 f/sec. Howfasttheneightof the shadow.hangingwhenthe personis 8 feetfromthe wall? Is the shadowincreasingruecreasingn heigntat thistime? Solution Let's start off with a sketch of this situation and the sketch here will ve similar to that of the previousproblem Thetop of the shadowwill be definedby the lightrays goingoverthe headof the personand so we againget yet anotherset of similartriangles Wall x' = 2.5 fi/sec Person y Spotlight 20- X X 20 In this casewe wantto determiney whenthe personis & ft fromwallorx -12 ft. Also if the personis movingtowardsthe wallat 2.5 ft/secthenthe personmustbe movingawayfromthe spotlightat 2.5 fv/sec andso we alsoknowthatx' = 2.5. In all the previousproblemsthatusedsimilartriangleswe usedthe similartrianglesto eliminate one of the variablesfromthe equationwe wereworkingwith In this case however, we can get the equationthatrelatesx andy directlyfromthe two similartriangles In this casethe equation we'regoingto workwithis, y 20 120 X X Mowall that we needto do is differentiatend plugvaluesintosolveto gety'. 120 120 y = -_ y = -12 (2.5) = -2.0833ft/sec The heightof the shadowis thendecreasingat a rate of 2.0833ft/sec. Okay, we've worked quite a few problems now that involved similar triangles in one form or another so thesekindso