AP®CALCULUS AB/CALCULUS BC2016 SCORING GUIDELINES© 2016 The College Board. Visit the College Board on the Web: www.collegeboard.org. Question 1 t(hours)0 1 3 6 8 Rt(liters / hour)13401190950740700Water is pumped into a tank at a rate modeled by 2Wt2000et20liters per hour for 0ddt8,wheretis measured in hours. Water is removed from the tank at a rate modeled by Rtliters per hour, whereRis differentiable and decreasing on 0ddt8.Selected values of Rtare shown in the table above. At time t0,there are 50,000 liters of water in the tank.(a)Estimate Rc2.Show the work that leads to your answer. Indicate units of measure.(b)Use a left Riemann sum with the four subintervals indicated by the table to estimate the total amount of water removed from the tank during the 8 hours. Is this an overestimate or an underestimate of the total amount of water removed? Give a reason for your answer.(c)Use your answer from part (b) to find an estimate of the total amount of water in the tank, to the nearest liter, at the end of 8 hours.(d)For 0ddt8,is there a timetwhen the rate at which water is pumped into the tank is the same as the rate at which water is removed from the tank? Explain why or why not.(a)231950 1190120 liters/hr31312RRR| c^ 1 : estimate2: 1 : units(b) The total amount of water removed is given by 80.Rtdt³801021332 6RtRRRRdt|³1 1340 2 1190 3 950 2 7408050 litersThis is an overestimate sinceRis a decreasing function. 1 : left Riemann sum3 : 1 : estimate 1 : overestimate with reason°®° ̄(c)80805050000 7836.Total 5000195325 8050 49786 li ers0ttdtW| |³^ 1 : integral2: 1 : estimate(d) WR0!00,WR880, and WtRtis continuous.Therefore, the Intermediate Value Theorem guarantees at least one timet, 0 t 8,for whichWtRt0, or WtRt.For this value oft, the rate at which water is pumped into the tank is the same as the rate at which water is removed from the tank.^ 1 : considers 2: 1 : answer with explanationWtRt
© 2016 The College Board.Visit the College Board on the Web: www.collegeboard.org.
© 2016 The College Board.Visit the College Board on the Web: www.collegeboard.org.
© 2016 The College Board.Visit the College Board on the Web: www.collegeboard.org.
© 2016 The College Board.Visit the College Board on the Web: www.collegeboard.org.
© 2016 The College Board.Visit the College Board on the Web: www.collegeboard.org.
