AP®CALCULUS AB/CALCULUS BC2015 SCORING GUIDELINESQuestion 1© 2015 The College Board. Visit the College Board on the Web: www.collegeboard.org. The rate at which rainwater flows into a drainpipe is modeled by the functionR, where 220sin35tRt§· ̈ ̧©¹cubic feet per hour,tis measured in hours, and 0dt8d.The pipe is partially blocked, allowing water to drain out the other end of the pipe at a rate modeled by Dt 0.04t320.4t0.96tcubic feet per hour, for 0ddt8.There are 30 cubic feet of water in the pipe at time t0.(a)How many cubic feet of rainwater flow into the pipe during the 8-hour time interval 0ddt8?(b) Is the amount of water in the pipe increasing or decreasing at time t3hours? Give a reason for your answer.(c)At what timet, 0ddt8,is the amount of water in the pipe at a minimum? Justify your answer.(d) The pipe can hold 50 cubic feet of water before overflowing. For t!8,water continues to flow into and out of the pipe at the given rates until the pipe begins to overflow. Write, but do not solve, an equation involving one or more integrals that gives the timewwhen the pipe will begin to overflow.(a)³8Rtdt76.5700^ 1 : integrand2: 1 : answer(b) RD3 30.313632 0Since RD3 3,the amount of water in the pipe is decreasing at time t3hours.3 and 3 1 : considers 2: 1 : answer and reasonRD® ̄(c)The amount of water in the pipe at timet, 0ddt8,ist30³>RxDx@dx.0RtDt0 t0, 3.271658tAmount of water in the pipe0 303.27165827.9645618 48.543686The amount of water in the pipe is a minimum at time t3.272(or 3.271) hours. 1 : considers 3 : 1 : answer 1 : justification0RtDt °®° ̄(d) ³w30>RtDt@dt500^ 1 : integral2: 1 : equation
©2015 The College Board.Visit the College Board on the Web: www.collegeboard.org.
©2015 The College Board.Visit the College Board on the Web: www.collegeboard.org.
©2015 The College Board.Visit the College Board on the Web: www.collegeboard.org.
©2015 The College Board.Visit the College Board on the Web: www.collegeboard.org.
©2015 The College Board.Visit the College Board on the Web: www.collegeboard.org.
