It appears that some crucial pieces of information are missing from your question that are necessary to solve for the linear depreciation model completely. Typically, in a linear depreciation model (which has the form y=mx+b), "y" represents the value of the home, "m" is the slope of the line which represents the rate of depreciation per year, "b" is the y-intercept which represents the initial value of the home, and "x" is the number of years since the start year (2002, in this case).
Let me guide you through each part, assuming we would have the necessary information:
(a) The slope of the line (m):
The slope of the line in a linear depreciation model reflects the rate of change in the value of the home per year. In other words, it represents the annual rate at which the home loses value, usually a negative number since the value is decreasing due to depreciation. Without concrete values, it's not possible to give a number here.
(b) The equation of the line:
The general form of the equation for a linear model is y = mx + b, where:
- m is the slope (negative for depreciation),
- x is the number of years since 2002,
- b is the initial value of the home in the year 2002 (in thousands of dollars),
- y is the value of the home after x years since 2002.
To create this equation, we need to know the initial value of the home in 2002 and the annual amount by which the home's value is decreasing.
For example, if the home was worth $200,000 in 2002 and it depreciates by $5,000 each year, the slope m would be -5 (as the value is in thousands), and b would be 200. So the equation would be y = -5x + 200.
(c) Predicting the value of the home in 2020:
To predict the value of the home in 2020, we would substitute x with the number of years since 2002, which is 2020 - 2002 = 18 years.
Using our example equation, y = -5x + 200, and substituting x with 18, we get:
y = -5(18) + 200
y = -90 + 200
y = 110
This would mean the predicted value of the home in 2020 is $110,000 (as the home value is represented in thousands).
Comparing this to the value predicted using an exponential model:
An exponential depreciation model would have a different form, typically y = a(1 - r)^t, where:
- a is the initial amount,
- r is the rate of depreciation (as a decimal),
- t is time in years since the initial year.
For such a model, the rate of depreciation affects the home's value exponentially, so the amount of depreciation changes each year, whereas, in a linear model, the amount depreciated each year is constant. This typically leads to different values for the projected home value, and the precise difference would depend on the rate r used in the exponential model. Without concrete values for the exponential model, we cannot make an accurate comparison here.
