Question in description

Question

At the start of the workday, 40,000 cubic feet of dirt was piled up at a construction site. Dirt will be added to the pile for the next 8 hours, satisfying the increasing differential equation  where the function P represents the total amount of dirt in the pile. P is measured in cubic feet, and t is measured in hours from the start of the workday.

Part A: Estimate the amount of dirt in the pile after 4 hours, using the tangent line to the graph of P at t = 0. (10 points)

Part B: Find and use  to determine if what you found in Part A was an underestimate or an overestimate at t = 4. (10 points)

Part C: Find the general solution to the differential equation  (10 points)

Answer 1

Part A: dP/(P - 6000) = 1/8dt; ∫dP/(P - 6000) = ∫1/8dt; ln(P - 6000) = 1/8t + lnC; P = 6000 + Ce^1/8t_;

At t = 0 40000 = 6000 + C; C = 34000; P(t) = 6000 + 34000e^1/8t

Tangent line equation at t = 0: P = 40000; dP/dt = 1/8(40000 - 6000) = 4250

So,P(t) = 6000 + 4250t; P(4) = 40000 + 4250·4 = 57000

Part B.: d²P/dt² = 1/8dP/dt = 1/8·4250 = 531.25 > 0. So, our result in Part A. was underestimate

Part C.: We get general solution in part A.:

dP/(P - 6000) = 1/8dt; ∫dP/(P - 6000) = ∫1/8dt; ln(P - 6000) = 1/8t + lnC; P = 6000 + Ce^1/8t_;

At t = 0 40000 = 6000 + C; C = 34000; P(t) = 6000 + 34000e^1/8t