Solving Steamboat Travel Time for Raft Drift

This problem requires calculating the time for a raft to drift downstream using given steamboat travel times.

Understanding River Speed Dynamics

Let:

• $D$ represent the distance between points A and B.
• $S_s$ represent the speed of the steamboat in still water.
• $S_c$ represent the speed of the river current.
• All times are measured in days.

Calculating Steamboat Speeds

The steamboat travels downstream (A to B) and upstream (B to A).

• Downstream Speed (A to B): The effective speed is the steamboat's speed plus the current's speed: $(S_s + S_c)$. The time taken is 5 days.
• Upstream Speed (B to A): The effective speed is the steamboat's speed minus the current's speed: $(S_s - S_c)$. The time taken is 7 days.

Using the relationship $Distance = Speed × Time$:

• Equation 1 (Downstream): $D = (S_s + S_c) × 5$
• Equation 2 (Upstream): $D = (S_s - S_c) × 7$

Deriving Current Speed

Rearranging the equations:

• From Equation 1: $D/5 = S_s + S_c$
• From Equation 2: $D/7 = S_s - S_c$

To find the current speed ($S_c$), subtract the second rearranged equation from the first:

$(D/5) - (D/7) = (S_s + S_c) - (S_s - S_c)$

$D * (1/5 - 1/7) = 2 * S_c$

Combine the fractions:

$D * ((7 - 5) / 35) = 2 * S_c$

$D * (2 / 35) = 2 * S_c$

Solve for $S_c$:

$S_c = D / 35$

Determining Raft Drift Time

A raft drifts only with the river current, so its speed is equal to $S_c$.

The time taken for the raft to travel distance $D$ is calculated as:

$Time_raft = Distance / Speed_raft$

$Time_raft = D / S_c$

Substitute the value of $S_c$ derived earlier:

$Time_raft = D / (D / 35)$

Simplify the expression:

$Time_raft = 35$

Thus, the raft will take 35 days to drift from A to B.