Solve Exponential and Logarithm Problems
Problems on Exponential and logarithm, some of the most important concepts in mathematics, are presented along with detailed solutions.
Problem 1:
The population P of a city increases according to the formula
\[ P = 5000 e^{a t}\]
where \( t \) is in years and \( t = 0 \) corresponds to 1980. In 1990, the population was 10000. Find the value of the constant a and approximate your answer to 3 decimal places.
Detailed Solution.
Problem 2:
The populations P1 and P2 of two cities are given by the formulas
\[ P1 = 10000 e^{k t} \]
\[ P2 = 20000 e^{ 0.01 t} \]
where \( k \) is a constant and t is the time in years with t = 0 corresponding to the year 2000. Find constant k so that the two populations are equal in the year 2020 and approximate your answer to 3 decimal places.
Detailed Solution.
Problem 3:
The level of sound \( D \) in decibels is defined as follows
\[
D = 10 \log\!\left(\dfrac{I}{10^{-16}}\right)
\]
where I is the sound intensity in watts per centimeters squared. Determine the level in decibels of a sound with intensity \(
I = 10^{-8}\ \text{watts/cm}^2 \)
.
Detailed Solution.
Problem 4:
Two sounds of intensities \(I_1\) and \( I_2 \) have decibel levels of 60 and 80 respectively. Use the formula for decibel level given in problem 3 to
determine the ratio of the intensities \( \dfrac{ I_2 }{ I_1} \)?
Detailed Solution.
Problem 5:
The spread of a virus through a city is modeled by the function
\[ N = \dfrac{15000}{1 + 100 e^{-0.5t}} \]
where \( N \) is the number of people infected by the virus after \( t \) days. How many days it takes for 2000 people of this city to be infected with the virus?
(approximate your answer to 3 decimal places).
Detailed Solution.
Problem 6:
The amount of a radioactive material decays according to the formula
\[ A(t) = A_0 e^{-k t} \]
where \( A_0 \) is the initial amount, \( k \) is a positive constant and \( t \) is the time in days. Find a formula for the half life of the material.
Detailed Solution.
Problem 7:
A logistic equation has the form
\[ y = \dfrac{a}{1 + b e^{-kt}}
\]
where\( a \), \( b \) and \(k \) are constants. Solve the above equation for \( t \).
Detailed Solution.
Problem 8:
The voltage \( V(t) \) across an electrical component is given by
\[ V(t) = V_0 e^{- a t } \]
where \( a \) is a positive constant depending on the values of the electronic components included in the electrical circuit and \( V_0 \) is the initial voltage.
Find \( t \) for which \( V(t) \) is equal to \( 50\% \) of \( V_0 \).
Detailed Solution.
Problem 9:
The signal ratio in decibels of an electronic system is given by the formula
\[ 10 \log\!\left(\dfrac{P_{o}}{P_{i}}\right) \]
where P o is the output power and Pi is the input power of the system. Find the input power \( P_i \) if the output power is equal to 10 mw and the signal ratio in decibels is equal to 10 decibels.
Detailed Solution.
Problem 10:
The current \( i \) flowing in an RL circuit is given by
\[
i = \dfrac{E}{R} \left[ 1 - e^{-\dfrac{Rt}{L}} \right]
\]
where \( E \) is the voltage applied to the circuit, \( R \) is the resistance and \( L \) is the inductance. Express \( i \) in terms of \( E \) and \( R \) when \( t = \dfrac{L}{R} \).
Detailed Solution.
More math problems with detailed solutions in this site.