The accompanying table shows the value of a car over time that was purchased for 16100 dollars, where x is years and y is the value of the car in dollars. Write an exponential regression equation for this set of data, rounding all coefficients to the nearest thousandth. Using this equation, determine the value of the car, to the nearest cent, after 16 years.
Answers
The value of car to the nearest cent, after 16 years will be $882.
Define the term exponential regression?It is a concept that describes processes that increase at a twice as fast rate. It is employed when the growth starts slowly but quickly accelerates without bounds or when the decay begins strongly but subsequently decelerates to zero.The preceding table displays the increase in value of a car that cost $16,100 over time, where x represents the number of years and y represents the car's cash value.
The formula for the exponential regression will be as follows:
y = A₀e∧(kx)
Where k is a constant and A₀ is the exponential regression coefficient.
So value of y would be $16,100 if x = 0. Next, we have
16,100 = A₀ e∧(k*0)
A₀ = 16,100
The equation will then be:
y = 16,100e∧(kx)
Hence value of y would be $13,431 if x = 1. Next, we have:
13,431 = 16,100e∧(k*1)
e∧(k) = 0.834
Considering all sides, we then have:
ln e∧(k) = ln 0.834
k = -0.181
The equation will then be:
y = 16,100e∧(-0.181*x)
For 16 years,
y = 16,100e∧(-0.181*16)
y = 16100*0.054
After that, a car's value will be:
y = 882
Thus, the value of the car to the nearest cent, after 16 years will be $882.
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