Respuesta :
Answer:
x≈6.4
Step-by-step explanation:
Exponential Functions:
y=ab^x
y=ab
x
a=\text{starting value = }24900
a=starting value = 24900
r=\text{rate = }6.75\% = 0.0675
r=rate = 6.75%=0.0675
\text{Exponential Decay:}
Exponential Decay:
b=1-r=1-0.0675=0.9325
b=1−r=1−0.0675=0.9325
\text{Write Exponential Function:}
Write Exponential Function:
y=24900(0.9325)^x
y=24900(0.9325)
x
Put it all together
\text{Plug in y-value:}
Plug in y-value:
15900=24900(0.9325)^x
15900=24900(0.9325)
x
\frac{15900}{24900}=\frac{24900(0.9325)^x}{24900}
24900
15900
=
24900
24900(0.9325)
x
Divide both sides by 24900
0.638554=0.9325^x
0.638554=0.9325
x
\log 0.638554=\log 0.9325^x
log0.638554=log0.9325
x
Take the log of both sides
\log 0.638554=x\log 0.9325
log0.638554=xlog0.9325
use power rule to bring x to the front
\frac{\log 0.638554}{\log 0.9325}=\frac{x\log 0.9325}{\log 0.9325}
log0.9325
log0.638554
=
log0.9325
xlog0.9325
Divide both sides by log(0.9325)
6.418279=x
6.418279=x
The value of the car will depreciate to $15900 in about 6.4 years.
What is exponential decay?
'Exponential decay describes the process of reducing an amount by a consistent percentage rate over a period of time.'
According to the given problem,
We know, exponential decay can be represented by formula,
y = [tex]a[/tex][tex](1 - b)^{x}[/tex]
Given, a = $24900
b = 6.75%
x = time period
y = $15900
⇒ [tex]15900 = 24900(1 - 0.0675)^{x}[/tex]
⇒ [tex]15900 = 24900(0.9325)^{x}[/tex]
⇒ [tex]\frac{15900}{24900}= 0.9325^{x}[/tex]
⇒ [tex]\frac{53}{83} = (0.9325)^{x}[/tex] [ Reducing L.H.S ]
Converting exponential to logarithm form,
⇒ x log (0.9325) = [tex]\frac{53}{83}[/tex]
⇒ x = [tex]\frac{53}{(83 )log(0.935)}[/tex]
⇒ x = 6.4 years
Hence, we can conclude, it will take 6.4 years to reduce the amount $24900 to $15900 by a rate of 6.75% per year.
Learn more about exponential decay here: https://brainly.com/question/27492127
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