Chapter 6.2, Problem 42E

To determine

To calculate: The value of x .

  

Answer to Problem 42E

The value of x is $98.1$ .

Explanation of Solution

Given information:

The figure,

  

Formula used:

The trigonometric ratios for a right angle triangle are defined as,

  $\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}},cos\theta =\frac{\text{adjacent}}{\text{hypotenuse}},\tan \theta =\frac{\text{opposite}}{\text{adjacent}},\csc \theta =\frac{\text{hypotenuse}}{\text{opposite}},\sec \theta =\frac{\text{hypotenuse}}{\text{adjacent}}$ and $\cot \theta =\frac{\text{adjacent}}{\text{opposite}}$ .

Calculation:

Consider the given figure,

Label the given figure as,

  

Recall that the trigonometric ratios for a right angle triangle are defined as,

  $\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}},cos\theta =\frac{\text{adjacent}}{\text{hypotenuse}},\tan \theta =\frac{\text{opposite}}{\text{adjacent}},\csc \theta =\frac{\text{hypotenuse}}{\text{opposite}},\sec \theta =\frac{\text{hypotenuse}}{\text{adjacent}}$ and $\cot \theta =\frac{\text{adjacent}}{\text{opposite}}$ .

Apply it, to estimate the value of trigonometric ratios,

In $△\text{ABC}$ , the value of tangent function is,

  $\begin{array}{c}\tan 30=\frac{\text{opposite}}{\text{adjacent}}\\ \tan 30=\frac{85}{\text{BC}}\end{array}$

Multiply both sides by $\text{BC}$ ,

  $\begin{array}{c}\tan 30=\frac{85}{\text{BC}}\\ \text{BC}\cdot \tan 30=\frac{85}{\text{BC}}\cdot \text{BC}\end{array}$

Simplify it with help of calculator,

  $\begin{array}{c}\text{BC}\cdot \tan 30=\frac{85}{\text{BC}}\cdot \text{BC}\\ \text{BC}\left(\frac{1}{\sqrt{3}}\right)=85\\ \text{BC}=85\sqrt{3}\\ \text{BC}=147.2\end{array}$

In $△\text{ABD}$ , the value of tangent function is,

  $\begin{array}{c}\tan 60=\frac{\text{opposite}}{\text{adjacent}}\\ \tan 60=\frac{85}{\text{BD}}\end{array}$

Multiply both sides by $\text{BD}$ ,

  $\begin{array}{c}\tan 60=\frac{85}{\text{BD}}\\ \text{BD}\cdot \tan 60=\frac{85}{\text{BD}}\cdot \text{BD}\end{array}$

Simplify it with help of calculator,

  $\begin{array}{c}\text{BD}\cdot \tan 60=\frac{85}{\text{BD}}\cdot \text{BD}\\ \text{BD}\left(\sqrt{3}\right)=85\\ \text{BD}=\frac{85}{\sqrt{3}}\\ \text{BD}=49.1\end{array}$

Now, the value of x will be estimated as the difference of BC and BD.

  $\begin{array}{c}x=\text{BC}-\text{BD}\\ x=147.2-49.1\\ x=98.1\end{array}$

Thus, the value of x is $98.1$ .