Calculus Writing Assignment: Modeling Using Differential Equations Each portion of each problem is worth 5 points. Total Points: 75 Answer each of the problems. Make sure to show all your work. Suppose the rate at which the volume in a tank decreases is proportional to the square root of the volume present. The tank initially contains 25 gallons, but has 20.25 gallons after 3 minutes. Answer the following. 1) Write a differential equation that models this situation. Let V represent the volume (in gallons) in the tank and t represent the time (in minutes). 2) Solve for the general solution (do not solve for V). 3) Use the initial condition to find the constant of integration, then write the particular solution (do not solve for V). 4) Use the second condition to find the constant of proportion. 5) Find the volume at t = 5 minutes. Round your answer to two decimal places. Suppose the cost of an object appreciates at a rate inversely proportional to the sum of its squared cost and 300. The object cost $240 when first purchased, but is worth $45 more after one year. Answer the following. 6) Write a differential equation that models this situation. Let c represent the cost (in dollars) of the object and t represent the time (in years). 7) Solve for the general solution (do not solve for c). 8) Use the initial condition to find the constant of integration, then write the particular solution (do not solve for c). 9) Use the second condition to find the constant of proportion. 10) Find the appreciation rate at t = 1 year. Round your answer to the nearest cent. The velocity of a particle varies directly as the product of its position and time squared. The particle has known positions s(0) = 3 and s(2) = 5. Answer the following. 11) Write a differential equation that models this situation. Let s represent the position of the particle and t represent the time. 12) Solve for the general solution. Write your answer as a function s(t). 13) Use the initial condition to find the constant of integration, then write the particular solution as a function s(t). 14) Use the second condition to find the constant of proportion. Round your answer to five decimal places. 15) Find the position of the particle at t = 3. Round your answer to three decimal places. Calculus Writing Assignment: Growth and Decay Models Each portion of each problem is worth 5 points. Total Points: 50 Answer each of the problems. Make sure to show all your work. The number of students at a school increases at a rate proportional to its current size. In 2005, the school had 1700 students. In 2009, the school had 1850 students. Answer the following. 1) Write a differential equation that models this situation. Let P represent the student population and let t represent the number of years since 2005. 2) Solve for the general solution. 3) Solve for the particular solution in terms of P and t (find the values of all constants). 4) Determine what year the school's population will reach 2100. 5) Determine the rate at which the population is increasing in 2010. Include units in your answer. The median value of a home in a particular market is decreasing exponentially. If the value of a home was initially $240,000, then its value two years later is $235,000. Answer the following. 6) Write a differential equation that models this situation. Let V represent the value of the home (in thousands of dollars) and t represent the number of years since its value was $240,000. 7) Solve for the particular solution in terms of V and t (find the values of all constants). 8) Determine when the value of the home will be 90% of its original value. 9) Determine the rate at which the value of the home is decreasing one year after it is valued at $235,000. Include units in your answer, and round the final value to the nearest dollar. 10) The relative rate of change in a quantity is defined as the rate of change for that quantity divided by the quantity present. Find the relative rate of change in the home's value at any time t. Calculus Writing Assignment: Separable Differential Equations Each portion of each problem is worth 5 points. Total Points: 75 Answer each of the problems. Make sure to show all your work. Suppose dy = 4 ye 5 x . Answer the following. dx 1) Separate the differential equation, then integrate both sides. 2) Write the general solution as a function y(x). Suppose dP = dt t and P(1) = 4. Answer the following. P 3) Separate the differential equation, then integrate both sides. 4) Find the constant of integration. 5) Write the particular solution as a function P(t). Suppose dy = y 5 cos sin5 . Answer the following. d 6) Separate the differential equation, then integrate both sides. 7) Write the general solution as a function y(). Suppose dn 1 = and n(1) = 2. Answer the following. dt 10n 8) Separate the differential equation, then integrate both sides. 9) Find the constant of integration. 10) Write the particular solution as a function n(t). Find the general solution to the following differential equations. Leave your answers in implicit form. 11) 3 t dy =8 dt 12) du x 1 + x 2 = dx tan u 13) dK = 6 2t + 3K Kt dt ( 2 dh z 1 + h 14) = dz 1+ z2 15) n 2 ) ds 3 6s 5 = 0 dn Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner