Chapter 8, Problem 24CT

To determine

To find: the value of x to the nearest integer

Answer to Problem 24CT

The value of x is $x=31$

Explanation of Solution

Given Information: Two right angled triangles with common altitude and some of the angles and sides being marked on the figure.

Formula used:

In a right triangle the sine of the angle is the ratio of the measure of side opposite to the angle to the hypotenuse.

  

It can be expressed as,

  $\sin B=\frac{a}{c}$

Similarly, the tangent of an angle is the ratio of opposite side to the adjacent side of the triangle expressed as,

  $\tan B=\frac{a}{b}$

Calculation: Consider the triangle given below.

  

Here, considering the triangle with acute angle 58⁰,

  $\tan 58°=\frac{z}{x}$ .

  $z=x\tan 58°$.......(1)

Now, considering the other right triangle,

  $\begin{array}{c}\sin 30°=\frac{z}{100}\\ z=100\sin 30°\\ =100\times \frac{1}{2}\\ =50\end{array}$

Now, substituting the value of z in (1) gives the value of x as,

  $\begin{array}{c}x=\frac{z}{\tan 58°}\\ =\frac{50}{\tan 58°}\end{array}$

Again , with the use of the table of trigonometric values in Page 311 in the textbook, the value of x can be obtained as,

  $\begin{array}{c}x=\frac{50}{1.6003}\\ =31.24\\ \simeq 31\end{array}$