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.

To solve this problem, we need to first determine the speed of the river's current by using the information provided about Sneha's journey. Then, we can use that current speed to find the total distance covered by Rita.

Let the speed of the river's current be $v$ km/h.

We are given that Sneha's speed in still water is 6 km/h. When rowing in a moving river, her effective speed changes depending on her direction:

$\text{Sneha's upstream speed}=6-v$

$\text{Sneha's downstream speed}=6+v$

Sneha rows a distance of 14 km upstream and 14 km downstream. The problem states that it takes her 48 minutes more to row upstream than downstream. First, let's convert 48 minutes into hours, because our speeds are in km/h:

$\frac{48}{60}=\frac{4}{5}\text{hours}$

Using the formula $\text{Time}=\frac{\text{Distance}}{\text{Speed}}$, we can write the time difference equation for Sneha:

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

To make this easier to solve, we can factor out the 14 from the numerators:

$14\left(\frac{1}{6-v}-\frac{1}{6+v}\right)=\frac{4}{5}$

Next, we find a common denominator for the terms inside the parentheses, which is $\left(6-v\right)\left(6+v\right)$, or $36-v^2$:

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

Simplifying the numerator by combining like terms:

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

This simplifies to:

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

We can divide both sides by 4 to simplify the equation further:

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

Cross-multiplying to eliminate the fractions gives:

$35v=36-v^2$

Rearranging the terms to form a standard quadratic equation:

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

We can factor this quadratic equation:

$\left(v+36\right)\left(v-1\right)=0$

This gives two possible solutions for the speed of the river: $v=-36$ or $v=1$. Since the speed of the river must be a positive value, we conclude that the river's velocity is 1 km/h.

Now, let's analyze Rita's journey. We know Rita's speed in still water is 5 km/h. With the river's velocity being 1 km/h, her effective speeds are:

$\text{Rita's upstream speed}=5-1=4\text{km/h}$

$\text{Rita's downstream speed}=5+1=6\text{km/h}$

Let the one-way distance Rita travels be $d$ km. The problem states that she makes a round trip (going upstream and then returning downstream) taking a total time of 100 minutes. We convert this total time to hours:

$\frac{100}{60}=\frac{5}{3}\text{hours}$

The sum of the time she spends going upstream and the time she spends going downstream must equal this total time. Setting up the equation:

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

To solve for $d$, we find a common denominator (12) for the fractions on the left side:

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

Combining the terms on the left side gives:

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

Multiplying both sides by $\frac{12}{5}$ to isolate $d$:

$d=4$

The one-way distance is 4 km. The question asks for the total distance Rita will cover, which is the sum of the upstream and downstream distances:

$\text{Total Distance}=d+d=4+4=8\text{km}$

Therefore, Rita covers a total distance of 8 km.