TG EAMCET 3 May 2025 Shift 2 Paper

Step 1: Find how far each ship goes in 2 hours.The first ship travels at 8 km per hour. In 2 hours, it travels:$8 \times 2 = 16\,\text{km}$The second ship travels at 12 km per hour. In 2 hours, it travels:$12 \times 2 = 24\,\text{km}$Step 2: Find the angle between the directions of the ships.The paths of the two ships make a $120^{\circ}$ angle with each other. We use this angle to find the distance between them after 2 hours.Step 3: Use the Cosine Law to find the distance between ships.The Cosine Law formula is: $c^2=a^2+b^2-2 a b \cos C$ where $a$ and $b$ are the distances traveled, and $C$ is the angle between the paths.Let $a=24\,\text{km},\ b=16\,\text{km}$, and $C=120^{\circ}$.Plug these values into the formula:$\cos 120^{\circ} = \frac{24^2+16^2-c^2}{2 \times 24 \times 16}$Since $\cos 120^{\circ} = -\frac{1}{2},\; -\frac{1}{2} \times 2 \times 24 \times 16 = 24^2+16^2-c^2$$24^2=576,16^2=256,\;\text{ and }\;2 \times 24 \times 16=768.\;\text{ So, }\;-\frac{1}{2} \times 768=576+256-c^2-384=832-c^2$Add $c^2$ on both sides and add 384 to both sides: $c^2=832+384=1216$Now, $1216=64 \times 19$. So, $c=8\sqrt{19}$So, the distance between the ships after 2 hours is $8\sqrt{19}\,\text{km}$.