Sediment buildup behind a dam can adversely affect a reservoir's storage capacity. Measurements of a reservoir in 1988 showed it had a capacity of 350,000 acre-feet of water. Measurements in 2018 showed that the reservoir's capacity has been decreasing at a rate of 2,500 acre-feet per year. Find a linear model of the number of acre-feet A of water in the reservoir n years after 1988.
To find the linear model, we first need to determine the equation of the line given the initial capacity in 1988 and the rate of decrease.
Let A represent the reservoir capacity in acre-feet and n represent the number of years after 1988.
The initial capacity in 1988 is 350,000 acre-feet, so this is the y-intercept of the line.
The rate of decrease is 2,500 acre-feet per year, so the slope of the line is -2,500.
Therefore, the linear model of the number of acre-feet A of water in the reservoir n years after 1988 is:
A = 350,000 - 2,500n
Use the model to predict the capacity of the reservoir in 2020.
To predict the capacity of the reservoir in 2020, we need to determine the number of years after 1988.
2020 is 32 years after 1988.
Substitute n = 32 into the linear model we found earlier:
A = 350,000 - 2,500(32)
A = 350,000 - 80,000
A = 270,000
Therefore, the predicted capacity of the reservoir in 2020 is 270,000 acre-feet.