please include all the steps necessary for each question

1)

Newton's Law of Cooling states that the rate of change of the temperature of an object. T, is proportional to the dirferenoe of Tand the temperature of its surrounding environment. A pot of chili with temperature 26C is placed into a 13C freezer. Aer 2 hours, the temperature of the chili is B\"C. Part A: Assuming the temperature Tot the chili follows Newton's Law of Cooling. write a differential equation for T. (10 points) Part B: What is the temperature Otlhe Chili aef 4 hours? (20 points) Part C: At What rm: I, will the Chili's temperature be -5"C? ('10 points) At the start of the workday, 30,000 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 OF = -(P - 8,000), where the function P represents the dt 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 2 hours, using the tangent line to the graph of Patt = 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 = 2. (10 points) it 2 Part C: Find the general solution to the differential equation OP _(P - 8,000). (10 points) itConsider the differential equation dy dx = 5(2x + 3) sin (x2 +3x + - ). Part A: Find the equation of the line tangent to the solution curve at the point (0, 5). (5 points) Part B: Find the second derivative at (0, 5) and use it to determine the concavity of the solution curve at that point. Explain. (10 points) Part C: Find the particular solution y = f(x) with initial condition f(0) = 5. (15 points)