Question

. Consumer Awareness In 1986, the average cost of a new midsize four-door sedan was $9000. In 1991, the average cost was $12,000. Determine an exponential model for predicting the cost of a new midsize car. Use the model to predict the cost of a comparable new car in the year 2000.

Answer 1

`\boxed{\boxed{y=9000(1.0592)^x}}`

And cost of a car in the year 2000 will be $20134.18

Solution-

Let's assume,

x = number of years after 1986

y = average cost of sedan in dollar

The exponential model that will model the scenario will be in the form of,

`y=ab^x`

where a and b are constants

As given that, in 1986 average cost of sedan was $9000 and in 1991 average cost of sedan was $12000

So, the points (0, 9000) and (5, 12000) will satisfy or lie on the exponential curve.

Putting (0, 9000) in the equation,

`\Rightarrow 9000=ab^0`

`\Rightarrow a* 1=9000`

`\Rightarrow a=9000`

Now, the equation becomes
`y=9000b^x`

Putting (5, 12000) in this equation,

`\Rightarrow 12000=9000b^5`

`\Rightarrow b^5=(12000)/(9000)`

`\Rightarrow b=\sqrt[5]{(4)/(3)}`

`\Rightarrow b=1.0592`

Putting the values,

`y=9000(1.0592)^x`

As we have to calculate the cost of sedan in 2000, so putting x=14(as 2000-1986=14),

`y=9000(1.0592)^(14)=20134.18`

Therefore, cost of a car in the year 2000 will be $20134.18