Chapter 1.8, Problem 69PS

Solution

To identify the correct answer: In which year the number of subscribers reach 50 million if there are S (in thousands) numbers of subscribers modeled by $S=858t^2+1412t+4982$ , t is the number of year since 1990.

The options are $\left(a\right)1991\left(b\right)1992\left(c\right)1996\left(b\right)2000$ .

The correct option is $\left(c\right)1996$ .

Given information:

The model is $S=858t^2+1412t+4982$ .

Explanation:

Write the given model is:

  $S=858t^2+1412t+4982$

Substitute 50000 for S into the equation,

  $50000=858t^2+1412t+4982$

Now subtract 50000 from both sides of the equation,

  $\begin{array}{c}50000-50000=858t^2+1412t+4982-50000\\ 0=858t^2+1412t-45018\end{array}$

Apply the quadratic formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ ,

  $\begin{array}{c}t=\frac{-1412\pm \sqrt{{\left(1412\right)}^2-4\left(858\right)\left(-45018\right)}}{2\left(858\right)}\\ t=\frac{-1412\pm \sqrt{1,993,774+154,501,776}}{1716}\\ t=\frac{-1412\pm \sqrt{156,495,520}}{1716}\\ t\approx 6\text{or}t\approx -8\end{array}$

Now add 6 years to 1990 to find the year in which number of subscribers will reach 50 million.

Therefore, the subscribers will reach 50 million in 1996.

The correct option should be option (c) 1996.