Physics to Go 1. Calculate the energy used, in joules, by each of the following: a) a $$ 1500-\mathrm{W} $$ hair dryer operating for 3 min b) a $$ 1200-\mathrm{W} $$ hair dryer operating for 4 min 2. If both situations described in Question 1 result in the same dryness of hair, which hair dryer is more efficient? 3. A 1200 -W hair dryer is used by several members of a family for a total of 30 min per day during a 30 -day month. How much electrical energy is consumed by the hair dryer during the month? Give your answer in: a) watt-hours b) kilowatt-hours 4. If the power company charges $$ \$ 0.15 $$ per $$ \mathrm{kW} \cdot \mathrm{h} $$ for electrical energy, what is the cost of using the hair dryer in Question 3 during the month? What is the cost for a year? 5. Not enough heat from the furnace reaches one bedroom in a home. The homeowner uses a portable 1350 -W electric heater 24 h per day to keep the bedroom warm during four cold winter months. At $$ \$ 0.12 $$ per kilowatt-hour, how much does it cost to operate the heater for the four months? (Assume two 30-day and two 31-day months.) 6. A portable CD player is rated at approximately 20 W and uses 4 AA batteries. a) Estimate the number of hours that you can listen to the music on a CD player before the batteries need replacing. b) Calculate the energy requirements of the CD player. c) Estimate the cost of 4 AA batteries. d) Calculate the cost per kilowatt-hour of a battery. e) Compare battery costs with the cost of electricity from the utilities (use approximately $$ \$ 0.10 $$ per kilowatt-hour).

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Sure, let's work through each of these physics problems step by step.

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## **1. Calculate the energy used, in joules, by each of the following:**

### **a) A $$ 1500\,\mathrm{W} $$ hair dryer operating for 3 minutes**

**Given:**
- Power ($$ P $$) = 1500 W
- Time ($$ t $$) = 3 minutes = 180 seconds

**Formula:**
$$ \text{Energy} (E) = P \times t $$

**Calculation:**
$$ E = 1500\,\mathrm{W} \times 180\,\mathrm{s} = 270,000\,\mathrm{J} $$

**Answer:**
The hair dryer uses **270,000 joules** of energy.

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### **b) A $$ 1200\,\mathrm{W} $$ hair dryer operating for 4 minutes**

**Given:**
- Power ($$ P $$) = 1200 W
- Time ($$ t $$) = 4 minutes = 240 seconds

**Formula:**
$$ \text{Energy} (E) = P \times t $$

**Calculation:**
$$ E = 1200\,\mathrm{W} \times 240\,\mathrm{s} = 288,000\,\mathrm{J} $$

**Answer:**
The hair dryer uses **288,000 joules** of energy.

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## **2. Efficiency Comparison of the Two Hair Dryers**

**Given:**
- Both hair dryers dry hair equally.
- Energy used:
  - 1500 W: 270,000 J
  - 1200 W: 288,000 J

**Analysis:**
Efficiency is defined as the usefulness of output per unit of energy input. Since both dryers achieve the same result (dryness) but the 1500 W dryer uses less energy:

**Conclusion:**
The **1500 W hair dryer is more efficient**, as it accomplishes the same task using less energy.

---

## **3. Electrical Energy Consumption of a 1200 W Hair Dryer Over a Month**

### **a) In Watt-hours (Wh)**

**Given:**
- Power ($$ P $$) = 1200 W
- Usage per day ($$ t $$) = 30 minutes = 0.5 hours
- Number of days = 30

**Formula:**
$$ \text{Energy} (E) = P \times t \times \text{days} $$

**Calculation:**
$$ E = 1200\,\mathrm{W} \times 0.5\,\mathrm{h} \times 30 = 18,000\,\mathrm{Wh} $$

**Answer:**
The hair dryer consumes **18,000 watt-hours** of electrical energy in the month.

### **b) In Kilowatt-hours (kWh)**

**Conversion:**
$$ 1\,\mathrm{kWh} = 1000\,\mathrm{Wh} $$

**Calculation:**
$$ E = \frac{18,000\,\mathrm{Wh}}{1000} = 18\,\mathrm{kWh} $$

**Answer:**
The energy consumption is **18 kilowatt-hours**.

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## **4. Cost of Using the Hair Dryer**

**Given:**
- Energy consumed = 18 kWh (from Problem 3)
- Cost per kWh = \$0.15

### **Monthly Cost**

**Formula:**
$$ \text{Cost} = \text{Energy} \times \text{Rate} $$

**Calculation:**
$$ \text{Cost}_{\text{month}} = 18\,\mathrm{kWh} \times \$0.15/\mathrm{kWh} = \$2.70 $$

### **Annual Cost**

**Given:**
- Monthly cost = \$2.70
- Months in a year = 12

**Calculation:**
$$ \text{Cost}_{\text{year}} = \$2.70 \times 12 = \$32.40 $$

**Answers:**
- **Monthly Cost:** \$2.70
- **Annual Cost:** \$32.40

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## **5. Operating Cost of a Portable Electric Heater Over Four Winter Months**

**Given:**
- Power ($$ P $$) = 1350 W
- Usage = 24 hours/day
- Duration = Four months (two 30-day and two 31-day months)
- Cost per kWh = \$0.12

**Calculations:**

### **Total Days:**
$$ 2 \times 30 + 2 \times 31 = 60 + 62 = 122\,\text{days} $$

### **Total Hours:**
$$ 24\,\text{hours/day} \times 122\,\text{days} = 2,928\,\text{hours} $$

### **Energy Consumed:**
Convert power to kilowatts:
$$ 1350\,\mathrm{W} = 1.35\,\mathrm{kW} $$

$$ E = 1.35\,\mathrm{kW} \times 2,928\,\mathrm{h} = 3,952.8\,\mathrm{kWh} $$

### **Cost:**
$$ \text{Cost} = 3,952.8\,\mathrm{kWh} \times \$0.12/\mathrm{kWh} = \$474.34 $$

**Answer:**
Operating the heater for four months costs **\$474.34**.

---

## **6. Analysis of a Portable CD Player Using 4 AA Batteries**

### **a) Estimated Listening Hours Before Batteries Need Replacing**

**Given:**
- Power ($$ P $$) = 20 W
- Number of AA batteries = 4

**Assumptions:**
- Each AA battery has a capacity of approximately 2,500 mAh at 1.5 V.

**Calculations:**

1. **Energy per Battery:**
   $$ E = V \times C = 1.5\,\mathrm{V} \times 2.5\,\mathrm{Ah} = 3.75\,\mathrm{Wh} $$

2. **Total Energy for 4 Batteries:**
   $$ E_{\text{total}} = 4 \times 3.75\,\mathrm{Wh} = 15\,\mathrm{Wh} $$

3. **Listening Time:**
   $$ t = \frac{E_{\text{total}}}{P} = \frac{15\,\mathrm{Wh}}{20\,\mathrm{W}} = 0.75\,\mathrm{hours} $$
   $$ 0.75\,\mathrm{hours} = 45\,\mathrm{minutes} $$

**Answer:**
You can listen to music for approximately **45 minutes** before needing to replace the batteries.

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### **b) Calculate the Energy Requirements of the CD Player**

**Given:**
- Power ($$ P $$) = 20 W
- Usage time from part a) = 0.75 hours

**Formula:**
$$ \text{Energy} (E) = P \times t $$

**Calculation:**
$$ E = 20\,\mathrm{W} \times 0.75\,\mathrm{h} = 15\,\mathrm{Wh} $$

**Answer:**
The CD player requires **15 watt-hours** of energy.

---

### **c) Estimate the Cost of 4 AA Batteries**

**Given:**
- Cost per AA battery ≈ \$0.50 (prices may vary)

**Calculation:**
$$ \text{Total Cost} = 4 \times \$0.50 = \$2.00 $$

**Answer:**
4 AA batteries cost approximately **\$2.00**.

---

### **d) Calculate the Cost per Kilowatt-hour of a Battery**

**Given:**
- Total energy from 4 batteries = 15 Wh
- Total cost = \$2.00

**Conversion:**
$$ 15\,\mathrm{Wh} = 0.015\,\mathrm{kWh} $$

**Formula:**
$$ \text{Cost per kWh} = \frac{\text{Total Cost}}{\text{Energy in kWh}} $$

**Calculation:**
$$ \text{Cost per kWh} = \frac{\$2.00}{0.015\,\mathrm{kWh}} = \$133.33/\mathrm{kWh} $$

**Answer:**
The cost per kilowatt-hour of the batteries is **\$133.33/kWh**.

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### **e) Compare Battery Costs with the Cost of Electricity from Utilities**

**Given:**
- Cost per kWh from utilities ≈ \$0.10

**Comparison:**
- **Battery Cost:** \$133.33/kWh
- **Utility Cost:** \$0.10/kWh

**Analysis:**
$$ \$133.33/\mathrm{kWh} \gg \$0.10/\mathrm{kWh} $$

**Conclusion:**
Using batteries to supply energy is **significantly more expensive** than using electricity from the grid. Specifically, batteries cost over **1,300 times more per kilowatt-hour** than utility electricity.

---

## **Summary**

- **Problem 1:** Calculated energy usage for hair dryers.
- **Problem 2:** Determined the more efficient hair dryer based on energy consumption.
- **Problem 3:** Computed monthly energy usage in Wh and kWh.
- **Problem 4:** Calculated both monthly and annual costs of operating the hair dryer.
- **Problem 5:** Evaluated the cost of operating an electric heater over four months.
- **Problem 6:** Analyzed the energy usage and cost-effectiveness of using batteries versus utility power for a CD player.

If you have any further questions or need additional explanations on any of these problems, feel free to ask!
**Simplified Answers:** 1. **Energy Used by Hair Dryers:** - 1500 W hair dryer for 3 minutes uses **270,000 joules**. - 1200 W hair dryer for 4 minutes uses **288,000 joules**. 2. **Efficiency Comparison:** - The 1500 W hair dryer is more efficient because it uses less energy to achieve the same dryness. 3. **Electrical Energy Consumption:** - 1200 W hair dryer used for 30 minutes per day over 30 days consumes **18,000 watt-hours** (18 kWh) monthly. 4. **Cost of Using the Hair Dryer:** - Monthly cost: \$2.70 - Annual cost: \$32.40 5. **Operating Cost of Electric Heater:** - Costs **\$474.34** to operate the heater for four months. 6. **Portable CD Player Analysis:** - Can listen for **45 minutes** on 4 AA batteries. - Total energy required: **15 watt-hours**. - Cost of batteries: **\$2.00**. - Cost per kilowatt-hour of batteries: **\$133.33/kWh**. - Batteries are **1,300 times more expensive** per kWh than utility electricity. If you have any more questions or need further clarification, feel free to ask!

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