author’s word

These math lessons has been written especially to meet the requirements of higher grade students. I’ve tried my best to present the work in a clear, simple and easy style so that students may not face any difficulty. Each lesson has solved examples and practice problems with answers.

While adding new topics is an ongoing process, efforts has been made to put the concepts in a logical sequence. In spite of my best efforts to make these lessons error free, some typing errors might have gone unnoticed. I shall be grateful to generous fellows if same are brought to my notice.

Worthy suggestions for improvement of these math lessons are always welcome.

Geometry

AP Calculus

Solved worksheets for AP calculus
- Worksheet 1 (Limits)
- Worksheet 2 (Differential Calculus)

Differential Equations

Vector Algebra

Algebra II

Three Dimensional Geometry

Trigonometry

Sequences and Series

Calculus

PreCalculus

Applications of exponential functions (Intensity of earthquakes and sound loudness)

Intensity of earthquakes and sound loudness

Richter scale is a logarithmic function to measure strength of earthquakes and its formula is given as,

$\dpi{150} \mathbf{R=log \left ( \frac{I}{I_{0}} \right )}$

Here R is the Richter scale magnitude of earthquake and $\dpi{120} \frac{I}{I_{0}}$ is the intensity ratio of earthquakes.

The intensity of earthquake will typically measure between 2 and 10 on Richter scale.

Example1: An earthquake is measured with a wave intensity 200 times as great as I₀. What is the magnitude of this earthquake using the Richter scale, to the nearest tenth?

Solution: Given that wave intensity 200 times as great as I₀

So $\dpi{120} I= 200I_{0}$

Substituting these values into the formula, we get

R = log(200) = 2.3

Magnitude of this earthquake is 2.3 on Richter scale.

Above formula can also be used in another form when Richter magnitude of two earthquakes is given and we need to compare their intensities.

$\dpi{150} \mathbf{\frac{I}{I_{0}}= 10^{M-M_{0}}}$

Where $\dpi{120} M_{0}$ is the magnitude of weaker earthquake and M is the magnitude of stronger earthquake.

Example2: A small tremor of magnitude 3.4 is followed by strong one of magnitude 4.1. How much stronger is the second tremor than the first one.

Solution: Here we need to find intensity of earthquakes.

Using formula $\dpi{120} \frac{I}{I_{0}}= 10^{M-M_{0}}$

$\dpi{120} \frac{I}{I_{0}}= 10^{4.1-3.4}$

$\dpi{120} \frac{I}{I_{0}}= 5.01$

Second tremor is 5 times stronger than the first one.

Example3: A strong earthquake with a magnitude of 5.7 is three times more intense than the weaker earthquake. Find the magnitude of weaker earthquake

Solution: Given that M =5.7 and $\dpi{120} I=3I_{0}$ . We need to find $\dpi{120} M_{0}$ .

$\dpi{120} \frac{3I_{0}}{I_{0}}=10^{5.7-M_{0}}$

$\dpi{120} 3=10^{5.7-M_{0}}$

$\dpi{120} log3=log10^{5.7-M_{0}}$

$\dpi{120} log3=(5.7-M_{0})log10$

$\dpi{120} 0.477 = 5.7-M_{0}$

$\dpi{120} M_{0}=5.02$

Formula for loudness of sound using sound intensity is :

$\dpi{150} \mathbf{dB-dB_{0}= 10log\left ( \frac{I}{I_{0}} \right )}$

Where dB is decibel value of loud sound and $\dpi{120} dB_{0}$ is decibel value of soft sound. $\dpi{120} \left ( \frac{I}{I_{0}} \right )$ is the intensity ratio of sounds.

Example4: A sound of 75dB is 50 times louder than a weaker sound. What is the loudness of weaker sound.

Solution: $dB-dB_{0}= 10log\left ( \frac{I}{I_{0}} \right )$

$75-dB_{0}= 10log\left ( \frac{50I_{0}}{I_{0}} \right )$

$75-dB_{0}= 10log\left ( \50 \right )$

$75-dB_{0}= 16.989$

$dB_{0}= 75-16.99= 58.01$

Example5: Two different sounds has their loudness level as 80 dB and 100dB. How much louder is the second sound?

Solution: $dB-dB_{0}= 10log\left ( \frac{I}{I_{0}} \right )$

$100-80= 10log\left ( \frac{I}{I_{0}} \right )$

$\frac{20}{10}= log\left ( \frac{I}{I_{0}} \right )$

$2= log\left ( \frac{I}{I_{0}} \right )$

$10^{2}=\frac{I}{I_{0}}$

$100 I_{0}=I$

So the louder sound is 100 times intense than the weaker sound.

Formula to find acid strength is:

$\dpi{150} \mathbf{pH=-log[H^{+}]}$

Where is the concentration of hydrogen ion and pH is acid strength(power of hydrogen)

Example6: Calculate the pH of the acid which has concentration of $\dpi{120} {\color{Red} H^{+}}$ ions as $\dpi{120} {\color{Red} 2.5\times 10^{-5}}$ .

Solution: $pH=-log[H^{+}]$

$pH=-log[2.5\times 10^{-5}]$

pH = 4.60

Example7: Find hydrogen concentration of acid which has a pH value of 3.7

Solution: $pH=-log[H^{+}]$

$3.7=-log[H^{+}]$

$-3.7=log[H^{+}]$

$10^{-3.7}=[H^{+}]$

$\dpi{120} 1.99\times 10^{-4} mol/L=\left [ H^{+} \right ]$

Practice problems:

A small tremor of magnitude 2.3 is followed by strong one of magnitude 5.3. How much stronger is the second tremor than the first one.
A strong earthquake with a magnitude of 5.9 is one thousand times more intense than the weaker earthquake. Find the magnitude of weaker earthquake.
A sound of 70dB is 8900 times louder than a weaker sound. What is the loudness of weaker sound.
Two different sounds has their loudness level as 97 dB and 110dB. How much louder is the second sound?
Find hydrogen concentration of acid which has a pH value of 8.7

Answers:

1000
2.9
30.5 dB
20
$\dpi{120} 2\times 10^{-9}$ mol/L

Facebook Comments

We have What You Need!

Celestial team is working hard to update content regularly, still if you feel any topic is left ,please do let us know. All valuable suggestions to make this site more meaningful and useful are appreciated.