Question: User: Phil is making a 40-kilometer canoe trip. If he travels at 30 kilometers per hour for the first 10 kilometers, and then at 15 kilometers per hour for the rest of the trip, how many minutes longer will it take him than if he makes the entire trip at 20 kilometers per hour? ANSWER WITH YES OR NO ONLY FIRST I HAVE TO KNOW HOW LONG WOUDL IT TAKE TO TRAVERSE 40 KILOMETERS DISTANCE USING THE TWO RATES YES OR NO
1. **Problem statement:** Phil makes a $40$-kilometer canoe trip. He goes the first $10$ kilometers at $30$ kilometers per hour, then the remaining $30$ kilometers at $15$ kilometers per hour. We want to know whether this trip takes longer than going the full $40$ kilometers at $20$ kilometers per hour.
2. **Formula used:**
- Time is found by $\text{time} = \frac{\text{distance}}{\text{speed}}$.
- For a trip with different speeds, find each part separately and add the times.
3. **Time for the first part:**
$$t_1 = \frac{10}{30}$$
Simplify:
$$t_1 = \frac{\cancel{10}}{\cancel{30}} = \frac{1}{3} \text{ hour}$$
4. **Time for the second part:**
$$t_2 = \frac{30}{15}$$
Simplify:
$$t_2 = \frac{\cancel{30}}{\cancel{15}} = 2 \text{ hours}$$
5. **Total time using the two rates:**
$$t = t_1 + t_2 = \frac{1}{3} + 2 = \frac{7}{3} \text{ hours}$$
6. **Time for the whole trip at $20$ kilometers per hour:**
$$t_{20} = \frac{40}{20} = 2 \text{ hours}$$
7. **Compare the times:**
$$\frac{7}{3} - 2 = \frac{7}{3} - \frac{6}{3} = \frac{1}{3} \text{ hour}$$
Convert to minutes:
$$\frac{1}{3} \times 60 = 20 \text{ minutes}$$
8. **Answer:** Yes, it takes longer. The trip using the two rates takes $20$ minutes longer than traveling the entire $40$ kilometers at $20$ kilometers per hour.