Domain, in yearly intervals, that contains all the years the car's value is less than $1,000 for the given exponential model is [ 14, ∞).
What is exponential model?
" Exponential model is defined as the curve which represents the growth or degradation of the model as per the given condition."
According to the question,
Given exponential model,
[tex]M(t) =20,000(\frac{4}{5} )^t[/tex]
M(t) = Midsize car's value
t = Time in years
Condition given to identify the domain for the given exponential model,
[tex]M(t) < 1000[/tex]
⇒[tex]20,000(\frac{4}{5} )^t < 1000[/tex]
⇒[tex](\frac{4}{5} )^t < \frac{1000}{20,000}[/tex][tex](\frac{4}{5} )^t < \frac{1}{20}[/tex]
Taking log both the sides we get,
⇒[tex]t ( log4-log5) < log1- log20[/tex]
⇒ [tex]t ( 0.6020 - 0.6989) = 1.3010[/tex]
⇒[tex]t(-0.0969 ) < (-1.3010)[/tex]
⇒[tex]t < \frac{1.3010}{0.0969}[/tex]
⇒[tex]t < 13.426[/tex]
After 13 years that is from 14 years to ∞ car's value is less than $1,000.
Therefore,
domain=[14,∞)
Hence, domain, in yearly intervals, that contains all the years the car's value is less than $1,000 for the given exponential model is [ 14, ∞).
Learn more about exponential model here
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