Chapter 7.4, Problem 39PS

Solution

To calculate : The correct option for total number of cassettes that will ship in $1994$ and after if the trend continues when the number of cassettes shipped in $1994$ was $345$ million. In each successive years, the number decreased by about $21.7\%$ .

Option (A): $420$ million

Option (B): $440$ million

Option (C): $615$ million

Option (D): $1.59$ billion

Option (D) is the correct option for total number of cassettes that will ship in $1994$ and after if the trend continues.

Given information :

The number of cassettes shipped in $1994$ was $345$ million. In each successive years, the number decreased by about $21.7\%$ .

Formula used : The sum of infinite geometric series with first term $a_1$ and common ration $r$ is calculated using the formula $S=\frac{a_1}{1-r}$ .

Here, $\left|r\right|<1$ . If $\left|r\right|\ge 1$ , then the series has no sum.

Calculation :

It can be observed that company’s profit has decreased by $21.7\%$ per year.

In the first year, the number of cassettes shipped in $1994$ was $345$ million

So,

  $a_1=345$

On the second year, profit has decreased by $21.7\%$ .

So,

  $\begin{array}{c}{a}_2=\left(1-0.217\right)a_1\\ =0.783a_1\end{array}$

Similarly,

  $a_3=0.783a_2$

It can be seen that; the sequence forms a geometric sequence with value of $r$ calculated as follows.

  $\begin{array}{c}r=\frac{a_2}{a_1}\\ =\frac{0.783a_1}{a_1}\\ =0.783\end{array}$

Observe all the terms as follows.

  $\begin{array}{c}{a}_2=0.783a_1\\ =0.783\left(345\right)\end{array}$

  $\begin{array}{c}{a}_3=0.783a_2\\ =0.783\cdot 0.783\left(345\right)\\ =345{\left(0.783\right)}^2\end{array}$

  $\begin{array}{c}{a}_4=0.783a_3\\ =0.783\cdot 345{\left(0.783\right)}^2\\ =345{\left(0.783\right)}^3\end{array}$

The general rule for the given geometric sequence is formed as $a_n=345{\left(0.783\right)}^{n-1}$ .

For the given series the value of $a_1$ is $345$ and $r$ is $0.783$ .

The sum of series can only exist if the common ratio is $\left|r\right|<1$ .

  $\begin{array}{c}\left|0.783\right|<1\\ 0.783<1\end{array}$

Then the sum of infinite geometric series is calculated using the formula $S=\frac{a_1}{1-r}$ .

  $\begin{array}{c}S=\frac{345}{1-0.783}\\ =\frac{345}{0.217}\\ \approx 1590\end{array}$

Thus, total number of cassettes that will ship in $1994$ and after if the trend continues is approximately $1.59$ billion.

So, the correct option is (D).