Chapter 6, Problem 19CR

Solution

(a)

To find: a function that represents the value y (in dollars) of the TV after t years.

  $y=1500{\left(0.86\right)}^t$ is the function.

Given: Initial value of TV is 1500 and decreases by 14% or $0.14$

Concept: exponential decay function, $y=a{\left(1-r\right)}^t$

Where, $\left(1-r\right)$ is the decay factor and r is the rate, a initial value and t is time.

Explanation: substitute given value in exponential decay function,

  $\begin{array}{l}y=1500{\left(1-0.14\right)}^t\\ =1500{\left(0.86\right)}^t\end{array}$

Therefore, the function is $y=1500{\left(0.86\right)}^t$ .

(b)

To find: the approximate monthly percentage decrease in value.

Monthly percentage decrease is $1.25\%$ .

Given: Initial value of TV is 1500 and decreases by 14% or $0.14$

Concept: exponential decay function, $y=a{\left(1-r\right)}^t$

Where, $\left(1-r\right)$ is the decay factor and r is the rate, a initial value and t is time.

Explanation: substitute given value in exponential decay function,

  $\begin{array}{l}y=1500{\left(1-0.14\right)}^t\\ =1500{\left(0.86\right)}^t\end{array}$

Now the value required in month, therefore, takes $t=\frac{1}{12}\left(12t\right)$ ,

  $y=1500{\left(0.86\right)}^{\frac{1}{12}\left(12t\right)}$

Use the power property simplify,

  $\begin{array}{l}y=1500{\left({0.86}^{\frac{1}{12}}\right)}^{\left(12t\right)}\\ =1500{\left(0.9875\right)}^{12t}\end{array}$

Therefore, the function is $y=1500{\left(0.9875\right)}^{12t}$ .

And the decay factor

  $\begin{array}{l}1-r=0.9875\\ 1-0.9875=r\\ 0.0125=r\end{array}$

Therefore monthly percentage decrease is $1.25\%$ .

(c)

To graph: the function and estimate the value of TV after 3 years.

Given: Initial value of TV is 1500 and decreases by 14% or $0.14$

Graph: Sketch the graph

  

Interpretation: Observation from the graph is at 3 year there y=950.

Therefore price after 3 year is approximate $950.