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

Answer-

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

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,

$\Rightarrow 9000=ab^0$

$\Rightarrow a\times 1=9000$

$\Rightarrow a=9000$

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

Putting (5, 12000) in this equation,

$\Rightarrow 12000=9000b^5$

$\Rightarrow b^5=\dfrac{12000}{9000}$

$\Rightarrow b=\sqrt[5]{\dfrac{4}{3}}$

$\Rightarrow b=1.0592$

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