AP CALCULUS WORD PROBLEM!

Question

at the beginning of 2010 a landfill contained 1400 tons of solid waste. The increasing 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 d^2W/dt^2 in terms of W. Use d^2W/dt^2 to determine whether your answer in part (a) is an underestimate or an overstatement of solid waste that the landfill contains at time t = 1/4.

Answer 1

a) (0,1400) in 2010

dW/dt = (1/25)[W-300]

use differentiation to get ∫ dW/(W-300) = ∫ (1/25)dt

ln|W-300| = (1/25)t + C     plug in (0,1400) to find C

ln (1400-300) = C  -->  C = ln (1100)

ln |W-300| = (1/25)t + ln (1100)

|W-300| = 1100e^((1/25)t)

W = 1100e^((1/25)t) + 300

plug in t= 1/4

W = 1100e^[(1/25)•(1/4)] + 300

W = 1100e^(1/100) + 300

W = 1411

 

b) use the second derivative which is (1/25)dW/dt to get

(1/25) [1/25(W-300)]

solve for W and you get 300

do a number line and you'll see that anything below 300 is negative (concave down) and anything above 300 is positive (concave up)

so the second derivative of 1/4 would be negative therefore it is an overapproximation