An environmental agency is working to reduce the amount of plastic that a community discards in the ocean. Currently, the community discards 6.2 milllion pounds of plastic annually, and the agency's goal is to eliminate that amount by collecting and recycling the plastic. If the agency incrases its collection and recycling capacity at a constant rate, and meets its goal at the end of the eight year, which of the following linear functions, f, could the agency use to model the amount of plastic being added to the ocean t years into the program ?

Options

A. `f (t) =- (62)/(40) t + 6.2`
B. `f (t) =- (31)/(40) t + 6.2`
C. `f (t) = (31)/(40) t + 6.2`
D. `f (t) = (62)/(40) t + 6.2`

Explanation

To solve the problem, we need to find a linear function that models the amount of plastic being added to the ocean \( t \) years into the program. Let's break this down step by step.

### Step 1: Understand the problem
The community currently discards 6.2 million pounds of plastic annually. The environmental agency aims to eliminate this amount by collecting and recycling plastic at a constant rate over 8 years.

### Step 2: Define the variables
Let:
- \( x \) = the amount of plastic (in million pounds) that the agency collects and recycles each year.
- \( t \) = the number of years into the program.

### Step 3: Determine the total amount collected by the end of year \( t \)
Since the agency increases its collection capacity at a constant rate, the total amount collected by the end of year \( t \) will be:
- Year 1: \( x \)
- Year 2: \( 2x \)
- Year 3: \( 3x \)
- ...
- Year \( t \): \( tx \)

Thus, at the end of year \( t \), the total amount collected is \( tx \) million pounds.

### Step 4: Set up the equation for year 8
By the end of the 8th year, the total amount collected should equal the amount discarded, which is 6.2 million pounds. Therefore, we have:
\[ 8x = 6.2 \]

### Step 5: Solve for \( x \)
To find \( x \), we rearrange the equation:
\[ x = \frac{6.2}{8} = 0.775 \text{ million pounds} \]

### Step 6: Write the function for plastic being added to the ocean
The amount of plastic being discarded each year is constant at 6.2 million pounds. The amount collected by the agency at year \( t \) is \( tx \). Therefore, the amount of plastic being added to the ocean at year \( t \) is:
\[ \text{Plastic added} = \text{Plastic discarded} - \text{Plastic collected} \]
\[ f(t) = 6.2 - tx \]
Substituting \( x \):
\[ f(t) = 6.2 - t(0.775) \]
This can be simplified to:
\[ f(t) = 6.2 - 0.775t \]

### Step 7: Finalize the function
To express it in a standard linear form:
\[ f(t) = -0.775t + 6.2 \]

### Conclusion
The function \( f(t) = -0.775t + 6.2 \) models the amount of plastic being added to the ocean \( t \) years into the program.

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