Assignment- Logarithms and Exponentials

1) The voltage across a capacitor in an RC circuit is given as

v(t) = 10 - 10e^-2t volts

a) Solve for the time (in seconds), t it takes to reach a voltage of v(t) = 9 volts.

b) Solve for the initial voltage. (i.e., the voltage when t is equal to zero)

c) Determine the voltage when t equals 0.01, 0.1, and 1 second(s).

d) Using the information from a)-d), plot the voltage as a function of time.

2) A water treatment plant tank has a drain hole that allows water to "leak" out. The height of the water as a function of time is given as

h(t) = h_oe^-(K/A)t

a) Using the properties of logs, solve for the time t at any height.

b) In terms of A and K, solve for the time t when the height of the water has drained by 50%. (i.e., h(t) = 0.50h_o)

3) The sound pressure level (measured in decibels, dBs) can be calculated using the following equation:

SPL = 20log (p/p_ref)

where p_ref is the threshold of hearing at 20 µPa. Determine the pressure that would result from a sound source with a level of:

a) 135 dBs (Threshold of Pain)

b) 120 dBs (Jackhammer)

c) 60 dBs (Speech)

d) 20 dBs (Leaves Rustling)

4) The amount of a radioactive isotope left after a given period of time is given as

A = A_o * 2^-(t/h)

where A_o is the original amount, h is the half-life of the isotope, and t is the time in hours.

a) If at 4 pm there are 2.5 grams and at 9 pm, there are 1.7 grams, what is the half life of the isotope?

b) How much will be left at 4 pm the next day?

5) Newton's Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature. As such, the temperature as a function of time (in minutes) can be calculated as

T(t) + T_a + (T_o - T_a)e^-kt

where T_a is the ambient temperature, T_o is the original temperature, and k is a constant.

a) Determine the time it takes a cup of hot chocolate (190 ?) to cool to 70 ? if it sits outside (20 ?) and has a k value of 0.08.

b) What k value would be needed to ensure the hot chocolate would take twice as long to cool to the same temperature?

6) Suppose your parents invest $1000 in a savings account for college at the time you are born. The average interest rate is 4% compounded quarterly (n = 4). Use the following equation:

A = P(1+(r/n))^nt

where P is the initial, principle amount of money, r is the interest rate, n is the number of times compounded in a year, and A is the amount after t time (in years).

a) How much money will be in the college account when you are 18 years old?

b) Tuition for one year at WSU is approximately $10,000. At the same interest rate, how much should your parents have initially invested to pay for one year of school?

c) At the same interest rate, how long would it take $1000 to grow to $10,000?