Chapter 13.3, Problem 56PPSE

Solution

To find: The central angle intercepted by the given arc.

Option B is correct.

Given:

Two arcs have same length. One arc intercepted by an angle $\frac{3\pi }{2}$ radians in a circle of radius 15 cm and the radius of the other circle is 25 cm.

Concept used:

The relation between the length of an arc $l$ , radius $r$ , and intercepted angle $\theta$ in a circle is $l=r\theta$ .

Calculation:

Given,

  ${\theta }_1=\frac{3\pi }{2},r_1=15,r_2=25$ .

Let $l_1$ and $l_2$ be the arcs in the two circles and ${\theta }_2$ be the arc intercepted angle in the second circle.

Since the two arcs have same length, therefore

  $\begin{array}{c}{l}_1=l_2\\ r_1{\theta }_1=r_2{\theta }_2\\ 15\times \frac{3\pi }{2}=25{\theta }_2\\ {\theta }_2=\frac{45\pi }{50}\\ {\theta }_2=\frac{9\pi }{10}\end{array}$

Therefore, the central angle intercepted by the arc is $\frac{9\pi }{10}$ radians.

Hence option B is correct.