The total time taken by a boat to cover 400 km downstream and 320 upstream is 40 hours. If downstream speed of the boat is 4 km/hr more than upstream speed of the boat, then find the time taken (in hours) by the boat to cover 720 km downstream.

Correct Answer: The correct option is 36 (which corresponds to the first option).

Step-by-Step Explanation:

Let us analyze the given information step-by-step to find the required time taken by the boat.

1. Define the Variables:

Let the speed of the boat in still water be $v$ km/hr.
Let the speed of the stream/river be $u$ km/hr.

Then:

Downstream speed ($D_s$) = $v+u$ km/hr
Upstream speed ($U_s$) = $v-u$ km/hr

2. Express the Given Relation Between Speeds:

The problem states that the downstream speed is 4 km/hr more than the upstream speed:

$D_s-U_s=4$

$(v+u)-(v-u)=4$

$2u=4\Rightarrow u=2$ km/hr

Now we can express upstream speed in terms of downstream speed:

$U_s=D_s-4$

3. Formulate the Equation for Total Time:

The total time taken to cover 400 km downstream and 320 km upstream is 40 hours.

$\text{Time}=\frac{\text{Distance}}{\text{Speed}}$

$\frac{400}{D_s}+\frac{320}{U_s}=40$

Substitute $U_s=D_s-4$ into the equation:

$\frac{400}{D_s}+\frac{320}{D_s-4}=40$

Divide the entire equation by 40 to simplify:

$\frac{10}{D_s}+\frac{8}{D_s-4}=1$

4. Solve for Downstream Speed ($D_s$):

Combine the fractions:

$\frac{10(D_s-4)+8D_s}{D_s(D_s-4)}=1$

$10D_s-40+8D_s=D_s^2-4D_s$

$18D_s-40=D_s^2-4D_s$

$D_s^2-22D_s+40=0$

Factorize the quadratic equation:

$D_s^2-20D_s-2D_s+40=0$

$D_s(D_s-20)-2(D_s-20)=0$

$(D_s-20)(D_s-2)=0$

Since downstream speed must be greater than 4 km/hr (as $U_s=D_s-4>0$), we reject $D_s=2$.

Thus, $D_s=20$ km/hr.

5. Calculate the Required Time:

We need to find the time taken to cover 720 km downstream:

$\text{Time}=\frac{\text{Distance}}{\text{Downstream Speed}}=\frac{720}{20}=36\text{hours}$

Hence, the time taken by the boat to cover 720 km downstream is 36 hours.