Rita and Sneha can row a boat at 5 km/h and 6 km/h in still water, respectively. In a river flowing with a constant velocity, Sneha takes 48 minutes more to row 14 km upstream than to row the same distance downstream. If Rita starts from a certain location in the river, and returns downstream to the same location, taking a total of 100 minutes, then the total distance, in km, Rita will cover is

The correct answer is 8.

Let us solve the problem step-by-step.

Step 1: Determine the speed of the river flow.

Let $v$ be the speed of the river current (in km/h).
We are given that Sneha's speed in still water is $v_S=6$ km/h.
Sneha's speed downstream = $6+v$ km/h.
Sneha's speed upstream = $6-v$ km/h.

Sneha travels a distance of $d=14$ km upstream and downstream.
Time taken upstream, $t_{\text{up}}=\frac{14}{6-v}$ hours.
Time taken downstream, $t_{\text{down}}=\frac{14}{6+v}$ hours.

We are given that the difference between the upstream time and downstream time is 48 minutes.
Converting 48 minutes into hours: 48 minutes = $\frac{48}{60}=\frac{4}{5}$ hours.

Setting up the equation:

$\frac{14}{6-v}-\frac{14}{6+v}=\frac{4}{5}$

Factor out 14 and combine the fractions on the left side:

$14\left(\frac{(6+v)-(6-v)}{36-v^2}\right)=\frac{4}{5}$

$14\left(\frac{2v}{36-v^2}\right)=\frac{4}{5}$

$\frac{28v}{36-v^2}=\frac{4}{5}$

Divide both sides by 4:

$\frac{7v}{36-v^2}=\frac{1}{5}$

Cross-multiplying gives:

$35v=36-v^2$

$v^2+35v-36=0$

Factoring the quadratic equation:

$(v+36)(v-1)=0$

Since speed must be positive, $v=1$ km/h.

Step 2: Calculate the distance Rita travels.

Rita's speed in still water is $v_R=5$ km/h.
Rita's speed downstream = $5+1=6$ km/h.
Rita's speed upstream = $5-1=4$ km/h.

Let $D$ be the one-way distance Rita rows.
Total time taken for the round trip is 100 minutes = $\frac{100}{60}=\frac{5}{3}$ hours.

The time equation for Rita is:

$\frac{D}{4}+\frac{D}{6}=\frac{5}{3}$

Combining the fractions on the left side with common denominator 12:

$\frac{3D+2D}{12}=\frac{5}{3}$

$\frac{5D}{12}=\frac{5}{3}$

Divide both sides by 5 and multiply by 12:

$D=\frac{12}{3}=4\text{km}$

The total distance covered by Rita (round trip, upstream + downstream) is:

$\text{Total Distance}=2\times D=2\times 4=8\text{km}$