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tcking3

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1b)

When a condenser discharges electricity, the instantaneous rate of change of the voltage is proportional to the voltage in the condenser. Suppose you have a discharging condenser and the instantaneous rate of change of the voltage is1/100of the voltage (in volts per second). How many seconds does it take for the voltage to decrease by 90 % ?

1c)

A radium sample weighs1gram at time t = 0. At time t = 10 years it has diminished to 0.997 grams. How long will it take to diminish to 0.5grams?

5d) Here is the solution of the differential equation:

y'[x] = (r y[x])/x2 with y[1] = a

Solution: a*er - r/x

Use derivative formulas and laws to explain this output.8) You put a tepid liquid refreshment into a very large refrigerator. The refrigerator is kept at a constant temperature S degrees. Let T[t] be the temperature of the liquid at time t minutes after the beverage is placed in the refrigerator. According to Newton's law of cooling, the rate of cooling T'[t] is proportional to the difference

T[t] - S ; that is, T'[t] = r (T[t] - S) = r T[t] - r S where r is a proportionality constant.

The following questions may be of some practical importance on a hot July afternoon. Suppose the refrigerator temperature S is held at 42 degrees (Fahrenheit), the original temperature of a bottle of a desirable beverage is 80 degrees , and the bottle of the beverage has cooled to 63 degrees after 10 minutes in the refrigerator. Plot the temperature of the bottle of the beverage as a function of time (in minutes). What will be the temperature of the bottle of the beverage after 30 minutes ? Approximately how many minutes will it take for the bottle of the beverage to cool to the refreshing temperature of 44 degrees ?