Chapter 4.1, Problem 51E

(a)

To determine

To write: a function in terms of x that models the value of the automobiles.

Answer to Problem 51E

  $V\left(x\right)=99000x^3+55000x^2+65000x$

Explanation of Solution

Given information: Classic cars: Sonia Orta invests in vintage automobiles. Three years ago, she purchased a 1953 Corvette roadster for $99,000. Two years ago, she purchased a 1929 Pierce-Arrow Model 125 for $55,000. A year ago she purchased a 1909 Cadillac Model Thirty for $65,000.

Calculation:

Given that the rate of appreciation is the same for all cars,

Then the values of the cars can be found using the

Compound interest, $A=P{\left(1+r\right)}^t:$

  $V\left(r\right)\$99000{\left(1+r\right)}^3+55000{\left(1+r\right)}^2+65000\left(1+r\right)$

Since $x=1+r,$ then:

  $V\left(x\right)=99000x^3+55000x^2+65000x$

(b)

To determine

the current value of the three automobiles, if the automobiles appreciate at an average annual rate of 15%.

Answer to Problem 51E

  $V\left(1.15\right)\approx \$298054.13$

Explanation of Solution

Given information: Classic cars: Sonia Orta invests in vintage automobiles. Three years ago, she purchased a 1953 Corvette roadster for $99,000. Two years ago, she purchased a 1929 Pierce-Arrow Model 125 for $55,000. A year ago she purchased a 1909 Cadillac Model Thirty for $65,000.

Calculation:

Given that the rate of appreciation is the same for all cars,

Then the values of the cars can be found using the

Compound interest, $A=P{\left(1+r\right)}^t:$

  $V\left(r\right)\$99000{\left(1+r\right)}^3+55000{\left(1+r\right)}^2+65000\left(1+r\right)$

Substitute $x=1+0.15=1.15$ & evaluate:

  $\begin{array}{l}V\left(1.15\right)=99000{\left(1.15\right)}^3+55000{\left(1.15\right)}^2+65000\left(1.15\right)\\ V\left(1.15\right)\approx \$298054.13\end{array}$