Question

let x be the number of years since 2000, let y be the amount of money left in the fund that year. Use a linear equation to model the amount of money left in the fund after so many years.A)B)C)

Answer 1

a. The company spends $22,000 per year. Then, the amount of money left, decrease by $22,000 per year. In terms of the linear model, this means that the slope of the line is -22,000.

Equation of a line in slope-intercept form

`y=mx+b`

where m is the slope and b is the y-intercept. In this case, the x-variable represents the number of years since 2000, and the y-variable represents the amount of money left in the fund.

In 2003, 3 years since 2000, there was still $852,000 left in the fund. This is equivalent to say when x = 3, y = 852,000. Substituting with these values and m = -22,000 into the equation, and solving for b:

`\begin{gathered} 852000=-22000\cdot3+b \\ 852000=-66000+b \\ 852000+66000=b \\ 918000=b \end{gathered}`

And the equation of the linear model is:

`y=-22000x+918000`

b. In the year 2009, x = 9. Substituting this value into the above equation and solving for y:

`\begin{gathered} y=-22000\cdot9+918000 \\ y=-198000+918000 \\ y=720000 \end{gathered}`

There was $720,000 left in the fund

c. If the fund is empty, then y = 0. Substituting this value into the model's equation and solving for x:

`\begin{gathered} 0=-22000x+918000 \\ 22000x=918000 \\ x=(918000)/(22000) \\ x\approx41.73 \end{gathered}`

At the end of the year 2041, the fund will be empty