9) You are planning a bicycle trip for which you want to average $$ 24 \mathrm{~km} / \mathrm{h} $$. You cover the first half of the trip at an average speed of $$ 21 \mathrm{~km} / \mathrm{h} $$. What must your average speed be in the second half of the trip to meet your goal? 10) You have 6.0 hours to travel a distance of 140 km by bicycle. How long will it take you to travel the first half at an average speed of $$ 21 \mathrm{~km} / \mathrm{h} $$ ? In the second half of the ride, you need to increase your average speed to make up for lost time. If you can maintain an average speed of $$ 25 \mathrm{~km} / \mathrm{h} $$, will you be able to reach your destination on time? $$ \square $$ What is the average speed you need to maintain in the second half of the bike ride to make up for lost time? $$ \square $$ $$ { }^{11)} $$ If light travels at $$ 3.00 imes 10^{8} \mathrm{~m} / \mathrm{s} $$, how long will it take light from the Sun to reach a planet that is 6.45 light years away? $$ \qquad $$ yr How far will the light have traveled in meters? (Use a value of exactly $$ \mathbf{3 6 5} $$ days for a year.) $$ \qquad $$ $$ imes 10^{16} \mathrm{~m} $$

Question

9) You are planning a bicycle trip for which you want to average $$ 24 \mathrm{~km} / \mathrm{h} $$. You cover the first half of the trip at an average speed of $$ 21 \mathrm{~km} / \mathrm{h} $$. What must your average speed be in the second half of the trip to meet your goal? 10) You have 6.0 hours to travel a distance of 140 km by bicycle. How long will it take you to travel the first half at an average speed of $$ 21 \mathrm{~km} / \mathrm{h} $$ ? In the second half of the ride, you need to increase your average speed to make up for lost time. If you can maintain an average speed of $$ 25 \mathrm{~km} / \mathrm{h} $$, will you be able to reach your destination on time? $$ \square $$ What is the average speed you need to maintain in the second half of the bike ride to make up for lost time? $$ \square $$ $$ { }^{11)} $$ If light travels at $$ 3.00 	imes 10^{8} \mathrm{~m} / \mathrm{s} $$, how long will it take light from the Sun to reach a planet that is 6.45 light years away? $$ \qquad $$ yr How far will the light have traveled in meters? (Use a value of exactly $$ \mathbf{3 6 5} $$ days for a year.) $$ \qquad $$ $$ 	imes 10^{16} \mathrm{~m} $$

UpStudy AI · Accepted Answer

Sure, let's tackle each of your questions step by step.

---

### **Problem 9**
**You are planning a bicycle trip for which you want to average $$ 24 \, \text{km/h} $$. You cover the first half of the trip at an average speed of $$ 21 \, \text{km/h} $$. What must your average speed be in the second half of the trip to meet your goal?**

**Solution:**

1. **Assume Total Distance $$ D $$:**
   - First half distance: $$ D/2 $$
   - Second half distance: $$ D/2 $$

2. **Time Taken for First Half:**
   $$
   t_1 = \frac{D/2}{21} = \frac{D}{42} \, \text{hours}
   $$

3. **Time Taken for Second Half at Speed $$ v $$:**
   $$
   t_2 = \frac{D/2}{v} = \frac{D}{2v} \, \text{hours}
   $$

4. **Total Time Desired for Entire Trip:**
   $$
   \text{Average Speed} = 24 \, \text{km/h} \Rightarrow \text{Total Time} = \frac{D}{24} \, \text{hours}
   $$

5. **Set Up the Equation:**
   $$
   t_1 + t_2 = \frac{D}{24} \
   \frac{D}{42} + \frac{D}{2v} = \frac{D}{24}
   $$

6. **Simplify the Equation:**
   $$
   \frac{1}{42} + \frac{1}{2v} = \frac{1}{24} \
   \frac{1}{2v} = \frac{1}{24} - \frac{1}{42} = \frac{7 - 4}{168} = \frac{3}{168} = \frac{1}{56}
   $$

7. **Solve for $$ v $$:**
   $$
   \frac{1}{2v} = \frac{1}{56} \Rightarrow 2v = 56 \Rightarrow v = 28 \, \text{km/h}
   $$

**Answer:**  
You must average **28  km/h** in the second half of the trip to meet your goal.

---

### **Problem 10**
**You have 6.0 hours to travel a distance of 140 km by bicycle.**

**a) How long will it take you to travel the first half at an average speed of $$21 \, \text{km/h}$$?**

**Solution:**
1. **First Half Distance:** $$ 140 \, \text{km} / 2 = 70 \, \text{km} $$
2. **Time for First Half:**
   $$
   t_1 = \frac{70}{21} \approx 3.333 \, \text{hours} = 3 \, \text{hours} \, 20 \, \text{minutes}
   $$

**Answer:**  
It will take you **3 hours and 20 minutes** to travel the first half.

---

**b) If you can maintain an average speed of $$25 \, \text{km/h}$$ in the second half, will you be able to reach your destination on time?**

**Solution:**
1. **Time Remaining:**
   $$
   t_{\text{remaining}} = 6.0 \, \text{hours} - 3.333 \, \text{hours} = 2.667 \, \text{hours} \, (2 \, \text{hours} \, 40 \, \text{minutes})
   $$
2. **Time to Cover Second Half at $$25 \, \text{km/h}$$:**
   $$
   t_2 = \frac{70}{25} = 2.8 \, \text{hours} \, (2 \, \text{hours} \, 48 \, \text{minutes})
   $$
3. **Total Time:**
   $$
   t_1 + t_2 = 3.333 + 2.8 = 6.133 \, \text{hours} > 6.0 \, \text{hours}
   $$

**Answer:**  
No, maintaining **25 km/h** in the second half will result in a total time of approximately **6 hours and 8 minutes**, which is **over the 6-hour limit**.

---

**c) What is the average speed you need to maintain in the second half of the bike ride to make up for lost time?**

**Solution:**
From **Problem 9**, we've determined that an average speed of **28 km/h** in the second half is required to achieve an overall average of **24 km/h** for the entire trip.

Alternatively, based on the time remaining:

1. **Required Time for Second Half:**
   $$
   t_2 = 6.0 \, \text{hours} - 3.333 \, \text{hours} = 2.667 \, \text{hours}
   $$
2. **Required Speed:**
   $$
   v = \frac{70}{2.667} \approx 26.25 \, \text{km/h}
   $$

**Answer:**  
You need to average approximately **26.25 km/h** in the second half to reach on time.

---

### **Problem 11**
**If light travels at $$3.00 \times 10^{8} \, \text{m/s}$$, how long will it take light from the Sun to reach a planet that is 6.45 light years away?**

**a) Time Taken**

**Solution:**
- **Definition:** 1 light-year is the distance light travels in one year.
- **Time to Travel 6.45 Light Years:**  
  $$ \text{Time} = 6.45 \, \text{years} $$

**Answer:**  
It will take **6.45 years** for light to reach the planet.

---

**b) How far will the light have traveled in meters? (Use a value of exactly $$365$$ days for a year.)**

**Solution:**
1. **Calculate Distance for One Light-Year:**
   $$
   \text{Distance} = \text{Speed} \times \text{Time} \
   = 3.00 \times 10^{8} \, \text{m/s} \times (365 \times 24 \times 3600 \, \text{s}) \
   = 3.00 \times 10^{8} \times 31,536,000 \, \text{s} \
   = 9.4608 \times 10^{15} \, \text{meters}
   $$
2. **Total Distance for 6.45 Light Years:**
   $$
   \text{Total Distance} = 6.45 \times 9.4608 \times 10^{15} \
   \approx 6.45 \times 9.4608 \times 10^{15} \
   \approx 6.10 \times 10^{16} \, \text{meters}
   $$

**Answer:**  
Light will have traveled approximately **6.10 × 10¹⁶ meters**.

---
**Problem 9:** To average 24 km/h for a trip, if the first half is at 21 km/h, the second half must be at **28 km/h**. **Problem 10:** - **a)** Traveling the first half at 21 km/h takes **3 hours and 20 minutes**. - **b)** Maintaining 25 km/h in the second half results in a total time of **6 hours and 8 minutes**, which is **over the 6-hour limit**. - **c)** To reach on time, you need to average approximately **26.25 km/h** in the second half. **Problem 11:** - **a)** It takes **6.45 years** for light to reach the planet. - **b)** Light travels approximately **6.10 × 10¹⁶ meters** in that time.

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