Chapter 18, Problem 42P

To determine

Find the percentage change in total amount of electricity generated in $2007$ compared to $1987$.

Answer to Problem 42P

The percentage change in total amount of electricity generated in $2007$ compared to $1987$ is $49.2\%$.

Explanation of Solution

Given data:

Refer to common Table for problems 18.36 through 18.42 as “Electricity Generation by fuel, 1980-2030 (billion kilowatt hours)-Data from U.S. Department of energy” in textbook,

The projected value of coal in the year $1980$ is $1161.562\text{billion kilowatt-hours}$.

The projected value of coal in the year $1990$ is $1594.011\text{billion kilowatt-hours}$.

The projected value of coal in the year $2005$ is $2040.913\text{billion kilowatt-hours}$.

The projected value of coal in the year $2010$ is $2217.555\text{billion kilowatt-hours}$.

The projected value of petroleum in the year $1980$ is $245.9942\text{billion kilowatt-hours}$.

The projected value of petroleum in the year $1990$ is $126.6211\text{billion kilowatt-hours}$.

The projected value of petroleum in the year $2005$ is $115.4264\text{billion kilowatt-hours}$.

The projected value of petroleum in the year $2010$ is $104.8182\text{billion kilowatt-hours}$.

The projected value of natural gas in the year $1980$ is $346.2399\text{billion kilowatt-hours}$.

The projected value of natural gas in the year $1990$ is $372.2652\text{billion kilowatt-hours}$.

The projected value of natural gas in the year $2005$ is $751.8189\text{billion kilowatt-hours}$.

The projected value of natural gas in the year $2010$ is $773.8234\text{billion kilowatt-hours}$.

The projected value of nuclear fuel in the year $1980$ is $251.1156\text{billion kilowatt-hours}$.

The projected value of nuclear fuel in the year $1990$ is $576.8617\text{billion kilowatt-hours}$.

The projected value of nuclear fuel in the year $2005$ is $774.0726\text{billion kilowatt-hours}$.

The projected value of nuclear fuel in the year $2010$ is $808.6948\text{billion kilowatt-hours}$.

The projected value of renewable in the year $1980$ is $284.6883\text{billion kilowatt-hours}$.

The projected value of renewable in the year $1990$ is $357.2381\text{billion kilowatt-hours}$.

The projected value of renewable in the year $2005$ is $375.8663\text{billion kilowatt-hours}$.

The projected value of renewable in the year $2010$ is $475.7432\text{billion kilowatt-hours}$.

Formula used:

Formula for the linear interpolation is,

$\frac{y-y_1}{x-x_1}=\frac{y_2-y_1}{x_2-x_1}$ (1)

Formula to calculate the total amount of electricity in the year $1980$ is,

$T_{1980}={\text{coal}}_{1980}+{\text{petroleum}}_{1980}+{\text{natural gas}}_{1980}+{\text{nuclear fuel}}_{1980}+{\text{renewable}}_{1980}$ (2)

Formula to calculate the total amount of electricity in the year $1990$ is,

$T_{1990}={\text{coal}}_{1990}+{\text{petroleum}}_{1990}+{\text{patural gas}}_{1990}+{\text{nuclear fuel}}_{1990}+{\text{renewable}}_{1990}$ (3)

Formula to calculate the total amount of electricity in the year $2005$ is,

$T_{2005}={\text{coal}}_{2005}+{\text{petroleum}}_{2005}+{\text{natural gas}}_{2005}+{\text{nuclear fuel}}_{2005}+{\text{renewable}}_{2005}$      (4)

Formula to calculate the total amount of electricity in the year $2010$ is,

$T_{2010}={\text{coal}}_{2010}+{\text{petroleum}}_{2010}+{\text{natural gas}}_{2010}+{\text{nuclear fuel}}_{2010}+{\text{renewable}}_{2010}$     (5)

Formula to calculate the percentage change of coal in $2007$ compared to $1987$ is,

$\%\text{change}=\frac{T_{2007}-T_{1987}}{T_{1987}}\times 100\%$ (6)

Calculation:

Substitute $1161.562\text{BkWh}$ for ${\text{coal}}_{1980}$, $245.9942\text{BkWh}$ for ${\text{petroleum}}_{1980}$, $346.2399\text{BkWh}$ for ${\text{natural gas}}_{1980}$, $251.1156\text{BkWh}$ for ${\text{nuclear fuel}}_{1980}$, and $284.6883\text{BkWh}$ for ${\text{renewable}}_{1980}$ in equation (1),

$\begin{array}{c}{T}_{1980}=\left[\begin{array}{c}\left(1161.562\text{BkWh}\right)+\left(245.9942\text{BkWh}\right)+\left(346.2399\text{BkWh}\right)\\ +\left(251.1156\text{BkWh}\right)+\left(284.6883\text{BkWh}\right)\end{array}\right]\\ =2289.6\text{BkWh}\end{array}$

Substitute $1594.011\text{BkWh}$ for ${\text{coal}}_{1990}$, $126.6211\text{BkWh}$ for ${\text{petroleum}}_{1990}$, $372.7652\text{BkWh}$ for ${\text{natural gas}}_{1990}$, $576.8617\text{BkWh}$ for ${\text{nuclear fuel}}_{1990}$, and $357.2381\text{BkWh}$ for ${\text{renewable}}_{1990}$ in equation (2),

$\begin{array}{c}{T}_{1990}=\left[\begin{array}{c}\left(1594.011\text{BkWh}\right)+\left(126.6211\text{BkWh}\right)+\left(372.7652\text{BkWh}\right)\\ +\left(576.8617\text{BkWh}\right)+\left(357.2381\text{BkWh}\right)\end{array}\right]\\ =3027.4971\text{BkWh}\end{array}$

Substitute $2040.913\text{BkWh}$ for ${\text{coal}}_{2005}$, $115.4264\text{BkWh}$ for ${\text{petroleum}}_{2005}$, $751.8189\text{BkWh}$ for ${\text{natural gas}}_{2005}$, $774.0726\text{BkWh}$ for ${\text{nuclear fuel}}_{2005}$, and $375.8663\text{BkWh}$ for ${\text{renewable}}_{2005}$ in equation (2),

$\begin{array}{c}{T}_{2005}=\left[\begin{array}{c}\left(2040.913\text{BkWh}\right)+\left(115.4264\text{BkWh}\right)+\left(751.8189\text{BkWh}\right)\\ +\left(774.0726\text{BkWh}\right)+\left(357.8663\text{BkWh}\right)\end{array}\right]\\ =4058.0972\text{BkWh}\end{array}$

Substitute $2217.555\text{BkWh}$ for ${\text{coal}}_{2010}$, $104.8182\text{BkWh}$ for ${\text{petroleum}}_{2010}$, $773.8234\text{BkWh}$ for ${\text{natural gas}}_{2010}$, $808.6948\text{BkWh}$ for ${\text{nuclear fuel}}_{2010}$, and $475.7432\text{BkWh}$ for ${\text{renewable}}_{2010}$ in equation (2),

$\begin{array}{c}{T}_{2010}=\left[\begin{array}{c}\left(2217.555\text{BkWh}\right)+\left(104.8182\text{BkWh}\right)+\left(773.8234\text{BkWh}\right)\\ +\left(808.6948\text{BkWh}\right)+\left(475.7432\text{BkWh}\right)\end{array}\right]\\ =4380.6346\text{BkWh}\end{array}$

To find the projected value total amount of electricity at the year $1987$ use the neighboring values of total amount of electricity generated at the year $1980$ and $1990$.

The diagrammatic representation for the given value is drawn below,

Substitute $1980$ for $x_1$, $1987$ for $x$, $1990$ for $x_2$, $2289.6\text{BkWh}$ for $y_1$,$T_{1987}$ for $y$ and $3027.4971\text{BkWh}$ for $y_2$ in equation (1),

$\begin{array}{c}\frac{T_{1987}-2289.6\text{BkWh}}{1987-1980}=\frac{3027.4971\text{BkWh}-2289.6\text{BkWh}}{1990-1980}\\ \frac{T_{1987}-2289.6\text{BkWh}}{7}=\frac{3027.4971\text{BkWh}-2289.6\text{BkWh}}{10}\\ \frac{T_{1987}-2289.6\text{BkWh}}{7}=\frac{737.8971\text{BkWh}}{10}\end{array}$ (7)

Equation (7) can be reduced as follows,

$\begin{array}{l}\frac{T_{1987}-2289.6\text{BkWh}}{7}=\frac{737.8971\text{BkWh}}{10}\otimes \\ T_{1987}-2289.6\text{BkWh}=\left(\frac{737.8971\text{BkWh}}{10}\right)\left(7\right)\\ T_{1987}-2289.6\text{BkWh}=516.52797\text{BkWh}\\ T_{1987}=\left(516.52797\text{BkWh}\right)\text{+}\left(2289.6\text{BkWh}\right)\\ T_{1987}=2806.1279\text{BkWh}\end{array}$

Therefore, the total amount of electricity generated in $1987$ is $T_{1987}=2806.1279\text{BkWh}$.

To find the projected value total amount of electricity at the year $2007$ use the neighboring values of total amount of electricity generated at the year $2005$ and $2010$.

The diagrammatic representation for the given value is drawn below,

Substitute $2005$ for $x_1$, $2007$ for $x$, $2010$ for $x_2$, $4058.0972\text{BkWh}$ for $y_1$,$T_{2007}$ for $y$ and $4380.6346\text{BkWh}$ for $y_2$ in equation (1),

$\begin{array}{c}\frac{T_{2007}-4058.0972\text{BkWh}}{2007-2005}=\frac{4380.6346\text{BkWh}-4058.0972\text{BkWh}}{2010-2005}\\ \frac{T_{2007}-4058.0972\text{BkWh}}{2}=\frac{4380.6346\text{BkWh}-4058.0972\text{BkWh}}{5}\\ \frac{T_{2007}-4058.0972\text{BkWh}}{2}=\frac{322.5374\text{BkWh}}{5}\end{array}$ (8)

Equation (8) can be reduced as follows,

$\begin{array}{l}\frac{T_{2007}-4058.0972\text{BkWh}}{2}=\frac{322.5374\text{BkWh}}{5}\\ T_{2007}-4058.0972\text{BkWh}=\left(\frac{322.5374\text{BkWh}}{5}\right)\left(2\right)\\ T_{2007}-4058.0972\text{BkWh}=129.01496\text{BkWh}\end{array}$

$\begin{array}{l}{T}_{2007}=\left(129.01496\text{BkWh}\right)\text{+}\left(4058.0972\text{BkWh}\right)\\ T_{2007}=4187.1121\text{BkWh}\end{array}$

Therefore, the total amount of electricity generated in $2007$ is $T_{2007}=4187.1121\text{BkWh}$.

Substitute $2806.1279\text{BkWh}$ for $T_{1987}$, and $4187.1121\text{BkWh}$ for $T_{2007}$ in equation (6),

$\begin{array}{c}\%\text{change}=\frac{4187.1121\text{BkWh}-2806.1279\text{BkWh}}{2806.1279\text{BkWh}}\times 100\%\\ =49.2\%\end{array}$

The percentage change in total amount of electricity generated in $2007$ compared to $1987$ is $49.2\%$.

Conclusion:

Thus, the percentage change in total amount of electricity generated in $2007$ compared to $1987$ is $49.2\%$.