Mariana is 10 years old and has grown 7 inches in 2 years. When she was 8 years old, she was 52 inches tall and began recording
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<h3>Given</h3>
The values of two houses (in thousands of dollars)
A) the nature of the function, linear or exponential, that can be used to model the value after x years
B) the actual function f(x) that can be used in each case
C) f(25) for each house. Is there a significant difference?
<h3>Solution</h3>A) The oddball numbers give you a clue immediately that the value of house 1 will be best modeled by an exponential function.
The value of house 2 is increasing steadily at 9,000 per year, so is modeled by a linear function.
B) The ratio of values from a given year to the year before for house 1 is
... 294.58/286 = 1.03
A check for other years reveals the same ratio, so the exponential function can be written for house 1 as
... f(x) = 286·1.03^x . . . . . value of house 1
In part A we determined the year-to-year difference in value for house 2 is 9,000. That is the slope of the linear function. Then (in thousands), that function is
... f(x) = 286 +9x . . . . . value of house 2
C) After 25 years, the house values are (in thousands of dollars)
The value of house 1 has more than doubled in the same time that the value of house 2 has increased by about 79%. This is a significant difference.
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An exponential function will always outperform a linear function over a long enough time period.