Question

There are two factories in a small town. Both of them emit carbon dioxide into the air. Factory 1 currently emits 120 tons per month, whereas factory 2 currently emits 180 tons per month. The technology of each factory is different, so their costs of reducing emissions are different as well. Factory 1's costs of reducing emissions is c₁ (x₁)=x₁² and factory 2's costs of reducing emissions is c₂ (x₂) = x₂²/4+20. Suppose the government in this town would like to cut monthly total emissions to half of the current level. To do that, the government has decided to impose a tax for every ton of pollution per month emitted by a factory. To achieve its desired goal, how would the government set the tax?

Answer 1

To achieve the goal of cutting monthly total emissions to half of the current level, the government should set a tax on carbon emissions that is at or above 18470 tons per month. This tax would make it more expensive for the factories to emit carbon dioxide than to reduce their emissions, encouraging them to take actions to lower their emissions.

Step-by-step explanation:

To achieve the goal of cutting monthly total emissions to half of the current level, the government needs to impose a tax on the factories. The tax should be set in a way that makes it more expensive for the factories to emit carbon dioxide than to reduce their emissions. In other words, the tax should be set at a level where the cost of reducing emissions becomes lower than the cost of emitting carbon dioxide.

Step-by-step explanation:

1. Create an equation for each factory's cost of reducing emissions:
   
   Factory 1: c₁(x₁) = x₁²
   
   Factory 2: c₂(x₂) = x₂²/4 + 20
2. Find the current emission levels for each factory:
   
   Factory 1: 120 tons per month
   
   Factory 2: 180 tons per month
3. Calculate the total emissions of both factories:
   
   Total emissions = 120 + 180 = 300 tons per month
4. Calculate the target emissions:
   
   Target emissions = Total emissions / 2 = 300 / 2 = 150 tons per month
5. Set up the following inequality to find the tax:
   
   c₁(x₁) + c₂(x₂) ≤ Tax
6. Substitute the emission levels and costs of reducing emissions into the inequality:
   
   120² + 180²/4 + 20 ≤ Tax
7. Simplify the inequality:
   
   14400 + 4050 + 20 ≤ Tax
   
   18470 ≤ Tax
8. Therefore, the government should set the tax at or above 18470 tons per month to achieve the goal of cutting monthly total emissions to half of the current level.