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# A lobster tank in a restaurant is 0.5 m long by 0.25 m wide by 40 cm deep. The force of the water (in Newtons) on the bottom of the tank = on each...

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1. A lobster tank in a restaurant is 0.5 m long by 0.25 m wide by 40 cm deep. The force of the water (in Newtons) on the bottom of the tank = on each of the larger sides of the tank = on each of the smaller sides of the tank = ( You need to compute the weight density of water to do this problem. The density of water is 1000 kg/m^{3} and the acceleration due to gravity is 9.8 m/s^2. Recall that weight is equal to mass times the acceleration due to gravity. ) 2. On August 12, 2000, the Russian submarine Kursk sank to the bottom of the sea, approximately 90 meters below the surface. Find the following at the depth of the Kursk: Water Pressure: (include units) The force (in Newtons) on a 5 meter square metal sheet held: Horizontally 90 meters below the surface = Vertically with its bottom 90 meters below the surface = (include units for each) ( You need to compute the weight density of water to do this problem. The density of water is 1000 kg/m^3 and the acceleration due to gravity is 9.8 m/s^2. Recall that weight is equal to mass times the acceleration due to gravity. ) 3. A trough is 2 feet long and 1 foot high. The vertical cross-section of the trough parallel to an end is shaped like the graph of y=x^{10} from x=-1 to x=1 . The trough is full of water. Find the amount of work in foot-pounds required to empty the trough by pumping the water over the top. Note: The weight of water is 62 pounds per cubic foot. 5. Rewrite the expression 5 \log x - 5 \log (x^2+1) +5 \log (x-1) as a single logarithm \log A. Then the function A= 6. The graphs of the functions y=a^x and y=\log_a x are symmetric with respect to the line y= 8. Scintillatium has a halflife of 20 minutes. If a sample has 800 grams, find a formula for its mass after t minutes. How much is left after 60 minutes? When will there be 50 grams left? 9. A bacteria culture starts with 900 bacteria and grows at a rate proportional
to its size. After 9 hours, there are 3000 bacteria. A. Find an expression for the number of bacteria after t hours. B. Find the number of bacteria after 10 hours. C. Find the growth rate after 10 hours. D. After how many hours will the population reach 30000? 10. Some time in the future a human colony is started on Mars. The colony begins with 50000 people and grows exponentially to 350000 in 200 years. Give a formula for the size of the human population on Mars as a function of t= time (in years) since the founding of the original colony Assuming the population continues to grow exponentially, how long will it take to reach a size of 700000? What is the rate of change of the size of the population 300 years after the founding of the original colony? 12. A cup of coffee at 200 degrees is poured into a mug and left in a room at 72 degrees. After 2 minutes, the coffee is 131 degrees. Assume that the differential equation describing Newton's Law of Cooling is (in this case) \frac{dT}{dt} = k(T-72). What is the temperature of the coffee after 12 minutes? After how many minutes will the coffee be 100 degrees? 13. A young person with no initial capital invests k dollars per year in a retirement account at an annual rate of return 0.04. Assume that investments are made continuously and that the return is compounded continuously. Determine a formula S(t) for the amount of money in the account at time t (this will involve the parameter k): S(t) = What value of k will provide 538000 dollars in 46 years? k = 14. Suppose that news spreads through a city of fixed size of 500000 people at a time rate proportional to the number of people who have not heard the news. (a.) Formulate a differential equation and initial condition for y(t), the number of people who have heard the news t days after it has happened. No one has heard the news at first, so y(0)=0. The "time rate of increase in the number of people who have heard the news is proportional to the number of people who have not heard the news" translates into the differential equation \frac{dy}{dt} = k ( ), where k is the proportionality constant. (b.) 6 days after a scandal in City Hall was reported, a poll showed that 250000
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