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Jack bought a new car in 2014 for 28,000. If the value of the car decreases by 14% each year, write an exponential model for the value of the car. Then estimate the year the car will have a value of 5,000

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3 votes

Answer: 1)
28000(0.86)^x

2) After 11.422 years( approx ) since 2014 the value of car will be 5000.

Explanation:

Since, the initial value of the car = 28000

The yearly decrease in the value of car = 14 %

Hence, the value of car after x years

=
28000(1-(14)/(100))^x

=
28000(1-0.14)^x

=
28000(0.86)^x

Which is the required exponential function that shows the value of car.

Now let after t years, the value of car is 5000,


\implies 28000(0.86)^y=5000


\implies 28(0.86)^y=5


\implies (0.86)^y = 0.178571429

By taking log both sides,


\implies y log(0.86)=log(0.178571429)


\implies y=11.4224479\approx 11.422\text{ years}

User Adam Trachtenberg
by
8.1k points
2 votes

Answer:

In 2026 car will have a value of $5,000.

Explanation:

We have been given that Jack bought a new car in 2014 for 28,000. If the value of the car decreases by 14% each year.

Since we know that an exponential function is in form:
y=a*b^x, where,

a = Initial value,

b = For decay or decrease b is in form (1-r), where r represents decay rate in decimal form.

Let us convert our given decay rate in decimal form.


14\%=(14)/(100)=0.14

Upon substituting a =28,000 and r=0.14 in exponential decay function we will get,


y=28,000(1-0.14)^x, where x represents number of years after 2014.

Therefore, the function
y=28,000(0.86)^x represents the value of car x years after 2014.

To find the number of years it will take to car have the value of $5,000, we will substitute y=5,000 in our function.


5,000=28,000(0.86)^x

Let us divide both sides of our equation by 28,000.


(5,000)/(28,000)=(28,000(0.86)^x)/(28,000)


0.1785714285714286=(0.86)^x

Let us take natural log of both sides of our equation.


ln(0.1785714285714286)=ln((0.86)^x)

Using natural log property
ln(a^b)=b*ln(a) we will get,


ln(0.1785714285714286)=x*ln(0.86)


(ln(0.1785714285714286))/(ln(0.86))=(x*ln(0.86))/(ln(0.86))


(-1.7227665977411033893)/(-0.1508228897345836)=x


x=11.422447\approx 12

As in the 12th year after 2014 car will have a value of $5,000, so we will add 12 to 2014 to find the year.


2014+12=2026

Therefore, in 2026 car will have a value of $5,000.

User Maplemaple
by
8.8k points

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