By Grace Ecko JojoPd 5,6

Ms. Zhao

2011 AP Calculus AB Free Response

#5Application of Derivative

What is a landfill exactly?A method of solid waste disposal in which refuse is buried between layers of dirt so as to fill in or reclaim low-lying ground.

Landfill Facts•Every year we fill enough garbage trucks to form a line that would stretch from the earth, halfway to the moon! (Just Live Green.com)•Only two human made structures are large enough to be seen from outer space: the Great Wall of China and the Fresh Kills Landfill, on the western shore of Staten Island!( Just Live Green.com)

•The total volume of solid waste produced in the U.S. each year is equal to the weight of more than 5,600 Nimitz Class air craft carriers, 247,000 space shuttles, or 2.3 million Boeing 747 jumbo jets (Beck).

AB/BC #5At the beginning of 2010, a landfill contained 1400 tons of solid waste. The increasing function W models the total amount of solid waste stored at the landfill. Planners estimate that W will satisfy the differential equation dW/dt=1/25 ( W-300) for the next 20 years. W is measured in tons, and t is measured in years from the start of 2010.

(a) Use the line tangent to the graph of W at t = 0 to approximate the amount of solid waste that the landfill contains at the end of the first 3 months of 2010 (time t=1/4)

(b) Find d²W/dt² in terms of W. Used d²W/dt² to determine whether your answer in part (a) is an underestimate or an overestimate of the amount of solid waste that the landfill contains at time t=1/4

AB/BC #5(c) Find the particular solution W = W(t) to the differential equation dW/dt= 1/25(W-300) with initial condition W(0) = 1400.

AB/BC #5(a) The line tangent to W is defined by the point (0,W(0))=(0,1400) and slope dW/dt=1/25(W-300)=44So the tangent is W-1400=44(t-0)→ W= 44t+1400 Thus, at t=1/4 W~44(.25)+1400=1411 tons There are 1411 tons of solid waste at the end of the first 3 months.

AB/BC #5(b) Take the derivative of dW/dt with respect to t :d²W/dt²=1/25(dW/dt)=1/25(1/25(W-300))=1/625(W-300)Since W is increasing for 0≤t≤20 , and W(O)=1400 , then d²W/dt² is positive on this interval. Hence, W is concave up on this interval. Thus the answer in part (a) is an underestimate of the solid waste in the landfill at t=1/4.

dW

dt

AB/BC #5(c) Separate the variables:dW/(W-300)=dt/25→lnǀW-300ǀ=t/25+CSince initial conditions are (0,1400) → lnǀ1400-300ǀ=0+C → C=ln1100→ lnǀW-300ǀ=t/25+ln1100Raise each side by e:→ W-300=e(t/25+ln1100)→ W(t)=1100e(t/25)+300

Change starts with YOU!REDUCE REUSE RECYCLE

•When grocery shopping go for goods that are not heavily packaged.•Buy less bottled water and use reusable container, like a thermos•Recycle worn out rechargeable batteries.

Citations•AP Central•Mr. Calculus•Just Live Green.com