ABC is a triangle with AB = 15 cm and AC = 17 cm. A semicircle is drawn with AC as diameter such that it passes through B. A person starting from A took ‘x’ minutes to reach C when following a semicircle path and took ‘y’ min when taking the triangle way (A → B → C) and took ‘z’ min when directly taking AC. Assuming the speed of the person as 1 cm/min. Find the value of \( \dfrac{14xy}{z}\).

Option 2 : 506

The correct answer is Option 2 i.e, __ 506__.

Given that the semicircle drawn with AC as diameter passing through B, It implies that ABC is a right angled triangle right angled at B. Thus by pythagoras theorem BC = 8 cm.

Given the value of v = speed of the person = 1 cm/min.

- Following the semicircular path, we get the value of x = \(\dfrac{\pi r}{v} = \dfrac{17\pi}{2}\) min.
- Following the triangular path, we get the value of y = \(\dfrac{(AB + BC)}{v} = 23\) min.
- Following directly the hypotenuse way, we get the value of z = \(\dfrac{AC}{v}=17\) min.

\(\therefore\) Required value =\(\displaystyle {14xy \over z} = {{14 \times {17π \over 2} \times 23} \over 17} = {7π \times 23} = {22 \times 23}=506\).