Suppose the government wants to impose a tax on fossil fuels to reduce carbon emissions. The cost benefit is modeled by $\ln ( 1 - \mathrm { P } ) = - 0.0031 - 0.0045 \mathrm { x }$, where $x$ represents the dollars of tax per ton of carbon emitted and $\mathrm { P }$ represents the percent reduction in emissions of carbon. ( $P$ is in decimal form.) Determine $P$ when $x = 44$. Round to three decimal places.
A) 1.818
B) -0.13
C) 0.182
D) -0.177

Question

Quizplus · Accepted Answer

To find $P$ when $x = 44$, substitute $x = 44$ into the given equation: $\ln(1 - P) = -0.0031 - 0.0045(44)$. Simplifying the right side gives $\ln(1 - P) = -0.0031 - 0.198$, which simplifies further to $\ln(1 - P) = -0.2011$. To solve for $P$, first exponentiate both sides to remove the natural logarithm, resulting in $1 - P = e^{-0.2011}$. Solving for $P$ gives $P = 1 - e^{-0.2011}$. Using a calculator, $P \approx 1 - 0.818 = 0.182$, rounded to three decimal places.

Suppose the Government Wants to Impose a Tax on Fossil ln⁡(1−P)=−0.0031−0.0045x\ln ( 1 - \mathrm { P } ) = - 0.0031 - 0.0045 \mathrm { x }ln(1−P)=−0.0031−0.0045x

Suppose the Government Wants to Impose a Tax on Fossil $\ln ( 1 - \mathrm { P } ) = - 0.0031 - 0.0045 \mathrm { x }$