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Suppose 100 grams of dye is dissolved in a 1000 liter tank. A mixture with a concentration of 3 grams/liter is being pumped into the tank at a rate of 4 liters per minute. As the fresh water is being added, a well mixed solution in the tank is being drained out at the same rate.  Solve the initial value problem to find the function Q(t) that gives the amount of dye in the mixture at time t. Submit your answer with your written work in “Final Exam PDF Paper Submission.”

The length of pregnancy is normally distributed with a mean of 270 days and standard deviation of 13. Find the following: The pregnancy lasts longer than 285 days. Is this unusually long?   The pregnancy lasts 254 days. Is this unusually low?  If the lowest 3% are considered unusually low, find the pregnancy length that separates unusually low from not. 

Use the following sample data set and solve for the variance (round to two decimal places): 71, 76, 78, 77, 86, 82, 88, 83, 73, 77, 80, 72, 79, 81, 89, 82

Students tracked how many hours they slept a night for a month. The mean was 8.37 with a standard deviation 1.84. Assume that the distribution is normal. Find the probabilities (round to three decimal places): Probability a student gets less than 6.75 hours of sleep. 

Students tracked how many hours they slept a night for a month. The mean was 8.37 with a standard deviation 1.84. Assume that the distribution is normal. Find the probabilities (round to three decimal places): Probability a student gets between 7 and 9 hours of sleep. 

Evaluate the following integral: (7 pts) Clearly show your work in the upload submission. (3 pts) Input your answer in the provided space below using the equation editor.

Find the volume of the solid bounded by the graphs of the given equations and revolved about the y-axis,    (5 pts) Sketch the typical rectangle that will be rotated, the solid, and a typical disk or washer in the upload submission. (4 pts) Show your work in the upload submission. Input the final answer in the given space below. (1 pt) The volume of the solid obtained by rotating the region about the specified line is

Determine the following limit: (6 pts) Clearly show your work in the upload submission. (2 pts) Input your answer in the provided space below.  

Evaluate the following integral. (8 pts) Clearly show your work in the upload submission. (2 pts) Input your answer in the provided space below. Type pi for π

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. (3 pts) Sketch the area enclosed by the given curves and draw a typical cylinder shell in the upload submission. (4 pts) Set up the integral for the volume based on your sketch and find the resulting volume in the upload submission. (2 pts) Input your answer with the correct unit in the provided space below. Type “pi” for “π” The exact volume is