Chapter 4.6, Problem 59PS

Solution

i.

To graph: The given function and determine the energy released by the indicated earthquake.

USA: $2.356\times {10}^4$ kilowatt-hours

GREECE: $1.145\times {10}^7$ kilowatt-hours

JAPAN: $1.2689\times {10}^8$ kilowatt-hours

Given:

A function $R=0.67log\left(0.37E\right)+1.46$.

Concept Used:

The intersection point on the graph represents the energy released by the earthquake. The $x$ axis of the graph represents the energy released in kilowatts hours and $y$ axis represents the Richter magnitude.

Calculation:

Substitute the function $R=0.67log\left(0.37E\right)+1.46$ in a graphing calculator to obtain the graph.

  

From the graph, it is understood that when the Richter magnitudes are $4.1,5.9$ and $6.6$ the corresponding released energy are $23555.7,11446300$ and $126894000$ kilowatt-hours respectively.

Conclusion:

The energy released in USA, GREECE and JAPAN are $2.356\times {10}^4$ , $1.145\times {10}^7$ and $1.2689\times {10}^8$ kilowatt-hours respectively.

ii.

To solve: The formed logarithmic equation to find the released energy.

USA: $2.354\times {10}^4$ kilowatt hours

GREECE: $1.142\times {10}^7$ kilowatt hours

JAPAN: $1.264\times {10}^8$ kilowatt hours

Given:

The Richter magnitudes are $4.1,5.9$ and $6.6$.

Concept Used:

Calculation:

According to the problem statement, following equation has been formed:

  $4.1=0.67\log \left(0.37E\right)+1.46$ ……(i)

  $5.9=0.67\log \left(0.37E\right)+1.46$ ……(ii)

  $6.6=0.67\log \left(0.37E\right)+1.46$ ……(iii)

Now solve equation (i) to obtain the energy released in the USA.

  $\begin{array}{c}4.1=0.67\log \left(0.37E\right)+1.46\\ 3.94=\log (0.37E)\\ {10}^{3.94}={10}^{\log \left(0.37E\right)}\end{array}$

Thus,

  $\begin{array}{c}0.37E=8709.63589\\ E=23539.55\\ E=2.354\times {10}^4\text{ kilowatt-hours}\end{array}$

Similarly, solve equation (ii) to obtain the energy released in the GREECE.

  $\begin{array}{c}5.9=0.67\log \left(0.37E\right)+1.46\\ 4.44=0.67\log \left(0.37E\right)\\ 6.626=\log \left(0.37E\right)\\ {10}^{6.626}={10}^{\log \left(0.37E\right)}\end{array}$

Thus,

  $\begin{array}{c}0.37E=4226686.14266\\ E=1.142\times {10}^7\text{ kilowatt-hours}\end{array}$

And finally, solve the equation (iii) to obtain the energy released in JAPAN.

  $\begin{array}{l}6.6=0.67\log \left(0.37E\right)+1.46\\ 5.14=0.67\log \left(0.37E\right)\\ 7.67=\log \left(0.37E\right)\\ {10}^{7.67}={10}^{\log \left(0.37E\right)}\end{array}$

Thus,

  $\begin{array}{c}0.37E=46773514.128\\ E=1.264\times {10}^8\text{ kilowatt-hours}\end{array}$

Conclusion:

The energy released in USA, GREECE and JAPAN is $2.354\times {10}^4$ , $1.142\times {10}^7$ and $1.264\times {10}^8$ kilowatt hours respectively.