Chapter 1.4, Problem 34E

Solution

To find: How much it takes time.

The time is equal to $1.62$ seconds.

Given information:

New Age Special Effects, Inc., prepares computer software based on specifications prepared by film directors. To simulate an approaching vehicle, they begin with a computer image of a $5-$ cm by $7-$ cm by $3-$ box. The program increases each dimension at rate of $2$ cm/sec.

Calculation:

The volume of the original box is $5\times 7\times 3=105$ cm³.

Assume that it takes $t$ time.

Since the rectangle increase at rate of $2$ km/sec, so the dimensions of the new box are $\left(5+2t\right),\left(7+2t\right)$ and $\left(3+2t\right)$ cm.

Since the volume of the new box is $5$ times its original size, so it can be expressed as below:

  $\left(5+2t\right)\times \left(7+2t\right)\times \left(3+2t\right)=5\times 105$

Solve for $t$ :

  $\begin{array}{c}\left(5+2t\right)\times \left(7+2t\right)\times \left(3+2t\right)=5\times 105\\ \left(35+14t+10t+4t^2\right)\times \left(3+2t\right)=525\\ \left(4t^2+24t+35\right)\times \left(3+2t\right)=525\\ 12t^2+72t+105+8t^3+48t^2+70t=525\\ 8t^3+60t^2+142t-420=0\\ t\approx 1.62\end{array}$

So, the time is $1.62$ seconds.

Hence, the time is $1.62$ seconds.