Chapter 3.8, Problem 14PSB

To determine

To calculate : The restrictions on the values of x .

Answer to Problem 14PSB

The restrictions on the values of x are $-12<x<-8$

Explanation of Solution

Given information : The following information has been provided

  $m\angle A>m\angle C$

  $\overrightarrow{AB}=4x+48$

  $\overrightarrow{BC}=8-x$

Formula used : The sides of a triangle are in the ratio of the angles opposite to them by sine’s law. Hence,

  $\frac{m\angle A}{\overrightarrow{BC}}=\frac{m\angle C}{\overrightarrow{AB}}$

Calculation : Now, we know that $m\angle A>m\angle C$ . So we can say that

  $\frac{m\angle A}{m\angle C}>1$

Now, substituting the value of $\frac{m\angle A}{m\angle C}$ from $\frac{m\angle A}{\overrightarrow{BC}}=\frac{m\angle C}{\overrightarrow{AB}}$ , we get

  $\frac{\overrightarrow{BC}}{\overrightarrow{AB}}>1$ thus, we get

  $\frac{8-x}{4x+48}>1$

Thus, solving it further, we get

  $x<-8$

But, we also know that sides f a triangle can’t be negative. Thus,

  $8-x>0$

  $4x+48>0$

Taking the intersection of all the above equations, we get

  $-12<x<-8$