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
[tex]y=mx+b[/tex]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:
[tex]\begin{gathered} 852000=-22000\cdot3+b \\ 852000=-66000+b \\ 852000+66000=b \\ 918000=b \end{gathered}[/tex]And the equation of the linear model is:
[tex]y=-22000x+918000[/tex]b. In the year 2009, x = 9. Substituting this value into the above equation and solving for y:
[tex]\begin{gathered} y=-22000\cdot9+918000 \\ y=-198000+918000 \\ y=720000 \end{gathered}[/tex]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:
[tex]\begin{gathered} 0=-22000x+918000 \\ 22000x=918000 \\ x=\frac{918000}{22000} \\ x\approx41.73 \end{gathered}[/tex]At the end of the year 2041, the fund will be empty
