A boat takes a total of 35 hours to cover 300 km downstream and 200 km upstream. The downstream speed of the boat is 10 km/hr more than its upstream speed. Find the time taken by the boat to cover 600 km downstream.

The correct option is 30 hours.

Step-by-step Explanation:

Let the upstream speed of the boat be $u$ km/hr.
Since the downstream speed of the boat is 10 km/hr more than its upstream speed, the downstream speed is:
$d=u+10$ km/hr.

We are given that the total time taken to cover 300 km downstream and 200 km upstream is 35 hours. Using the relationship $\text{Time}=\frac{\text{Distance}}{\text{Speed}}$, we can write the equation:
$\frac{300}{u+10}+\frac{200}{u}=35$

To simplify the equation, we can divide the entire equation by 5:
$\frac{60}{u+10}+\frac{40}{u}=7$

Now, find a common denominator and combine the terms on the left side:
$\frac{60u+40(u+10)}{u(u+10)}=7$

Expand the numerator:
$\frac{60u+40u+400}{u^2+10u}=7$
$\frac{100u+400}{u^2+10u}=7$

Cross-multiply to form a quadratic equation:
$100u+400=7(u^2+10u)$
$100u+400=7u^2+70u$

Rearrange the terms into standard quadratic form ($a{u}^2+bu+c=0$):
$7u^2-30u-400=0$

We can solve this quadratic equation using the quadratic formula:
$u=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$
Here, $a=7$, $b=-30$, and $c=-400$.
$u=\frac{-(-30)\pm \sqrt{{(-30)}^2-4\left(7\right)(-400)}}{2\left(7\right)}$
$u=\frac{30\pm \sqrt{900+11200}}{14}$
$u=\frac{30\pm \sqrt{12100}}{14}$
$u=\frac{30\pm 110}{14}$

Since speed must be a positive value, we take the positive root:
$u=\frac{30+110}{14}=\frac{140}{14}=10$ km/hr.

Therefore, the upstream speed of the boat is 10 km/hr.
The downstream speed of the boat is:
$d=u+10=10+10=20$ km/hr.

Now, we find the time taken to cover 600 km downstream:
$\text{Time}=\frac{\text{Distance}}{\text{Downstream Speed}}=\frac{600}{20}=30$ hours.