72472520_726456917870695_6592453961008021504_n.jpg - CHAPTER 1 FIRST-ORDER DIFFERENTIAL EQUATIONS AND THEIR APPLICATION 19 b Solve the differential

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Unformatted text preview: CHAPTER 1: FIRST-ORDER DIFFERENTIAL EQUATIONS AND THEIR APPLICATION 19 b) Solve the differential equation and deter- mine the amount in the account after 10 yr. 30. An amount of $10,000 is invested at 12% an- c) How much more would the account have nual interest compounded daily. An additional after 10 yr if the full amount earned investment of $B is made daily. What should interest? B be in order for the investment account to be $100,000 after 10 yr? 28. An amount of $10,000 is deposited in a bank daily. that pays 9% annual interest compounded 31. An amount of $100 is deposited in a foreign bank that pays 20% annual interest com- pounded daily. Each day you make a transac- a) If you withdraw $10 a day, how much money tion of amount f() dollars. Part of the year do you have after 3 yr? you are able to deposit money [f(t) > 0], b) How much can you withdraw each day and part of the year you make withdrawals if the account is to be depleted in exactly [f(1) < 0]. If / is in years, these transactions 10 yr? occur in the following cyclical yearly pattern: 29. A amount of $1000 is deposited in a bank that pays 8% annual interest compounded daily. A f ( 1 ) = 400 365 cos (2+1) $/day deposit of B dollars is made daily. = 400 cos (27rt) $/year. a) What should B be in order to have $10,000 after 5 yr? a) Find the amount of money in the account b) Determine the function B(x) that gives the as a function of t. daily deposit needed to have $10,000 after x years. [ B(5) is computed in part (a).] b) Graph your answer to (a) for 10 yr (a sketch will do if no computer is available). 1.9 Mixture Problems In this section, a quantity Q(), such as the amount of some pollutant in a water tank, varies with time. Further amounts of this quantity are being added. The addition will be called inflow. Simultaneously, some of this quantity is being lost. The quantity lost will be called the outflow. In the case of a water tank, the loss could be due to evaporation, overflow, an open valve, or all three. In all cases we assume that the tank is very well mixed (through fast stirring if necessary), so that the concentration of the pollutant will be assumed to be the same throughout all portions of the tank. The beginning idea for analyzing these problems is the fundamental physical principle of conservation of the quantity Q: Rate of change inflow rate outflow rate of Q dt. of Q of Q (1) where we have noted that dQ/di, the time derivative of Q, is the rate of change of Q with respect to time t....
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