TIM PADDLED HIS KAYAK 12 KM UPSTREAM AGAINST A 3 KM/H CURRENT AND BACK AGAIN IN 5 HR 20 MINUTES. iN THAT TIME
HOW FAR COULD HE HAVE PADDLED IN STILL WATER
12/(v+3) + 12/(v-3) = 5.33 hours
Solve for this speed in still water, v.
12(v-3) + 12(v+3) = 5.33 (v^2 -9)
24 v = 5.33 v^2 - 48
Use the quadratic equation and take the positive root for v.
v = 6.0 km/h
In the same time, the distance he could travel in still water is
D = 5.33 * v = 32 km
To solve this problem, we can use the concept of relative velocity. Let's break it down step by step:
1. The first thing we need to do is set up an equation using the formula: time = distance / speed.
2. TIM paddled his kayak 12 km upstream against a 3 km/h current. The total time taken for this part of the journey can be represented as: 12 / (v + 3), where v is the speed of TIM in still water.
3. TIM also paddled 12 km downstream with the same 3 km/h current. The total time taken for this part of the journey can be represented as: 12 / (v - 3).
4. We are given that the total time for the entire journey is 5 hours and 20 minutes. We need to convert this to hours, which is 5 + (20 / 60) = 5.33 hours.
5. Now, we can set up our equation: 12 / (v + 3) + 12 / (v - 3) = 5.33.
6. Next, we simplify the equation by multiplying through by the common denominator (v + 3)(v - 3):
12(v - 3) + 12(v + 3) = 5.33(v^2 - 9).
7. Expanding and simplifying the equation, we get:
24v = 5.33v^2 - 48.
8. Rearranging the equation, we have:
5.33v^2 - 24v - 48 = 0.
9. At this point, we have a quadratic equation. We can solve it using the quadratic formula:
v = (-b ± √(b^2 - 4ac)) / (2a),
where a = 5.33, b = -24, and c = -48.
10. Plugging in these values and solving, we get two possible solutions for v: v ≈ 6.22 km/h and v ≈ -1.44 km/h. Since we are talking about the speed of a kayak, we discard the negative value.
11. Therefore, the speed of TIM in still water is approximately 6.22 km/h.
12. Finally, to find the distance TIM could have paddled in still water in the given time, we multiply the speed by the total time: distance = 5.33 * 6.22 ≈ 32 km.
So, TIM could have paddled approximately 32 km in still water in the given time.