Question

Two ships A and B are sailing straight away from the fixed point O along the routes such that $\angle AOB={60}^{\circ }$ always.At a certain instance, OA = 4 km, OB = 3km and ship A is sailing at the rate of 20km/hr and ship B is sailing at the rate of 30km/hr. Then the distance between A and B is changing at the rate (in km/hr) :

• A. $\frac{18}{\sqrt{13}}$
• B. $\frac{160}{\sqrt{13}}$
• C. $\frac{80}{\sqrt{13}}$
• D. $\frac{40}{\sqrt{13}}$

Answer 1

The correct option is C $\frac{80}{\sqrt{13}}$

Using cosine rule,
$S=\sqrt{x^2+y^2-xy}Givenx=4,y=3km\frac{dx}{dt}=20km/hr,\frac{dy}{dt}=30km/hr\frac{dS}{dt}=\frac{1}{2\sqrt{x^2+y^2-xy}}[2x\frac{dx}{dt}+2y\frac{dy}{dt}-(x\frac{dy}{dt}+y\frac{dx}{dt})]=\frac{1}{2\sqrt{x^2+y^2-4\times 3}}[2\times 20\times 4+2\times 3\times 30-(4\times 30+3\times 20\left)\right]=\frac{80}{\sqrt{13}}$