A small heater is used to keep critical components from getting too cold in space. The device yields an output of 87.5 kilocalories [kcal] of heat energy over a time span of 2 hours [h]. The circuit contains a 0.4 ohm [2] resistor and is 90% efficient. What is the input voltage in units of volts [V]?
A small heater is used to keep critical components from getting too cold in space. The device yields an output of 87.5 kilocalories [kcal] of heat energy over a time span of 2 hours [h]. The circuit contains a 0.4 ohm [2] resistor and is 90% efficient. What is the input voltage in units of volts [V]?
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
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Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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![**Problem: Calculating Input Voltage for a Space Heater Circuit**
A small heater is used to keep critical components from getting too cold in space. The device yields an output of 87.5 kilocalories (kcal) of heat energy over a time span of 2 hours. The circuit contains a 0.4 ohm (Ω) resistor and is 90% efficient. What is the input voltage in units of volts (V)?
---
**Solution Explanation:**
1. **Energy Conversion:**
- To find the power requirement in watts (W), convert kilocalories to joules:
\[
\text{1 kcal} = 4184 \text{ joules}
\]
\[
\text{87.5 kcal} = 87.5 \times 4184 \text{ joules}
\]
2. **Calculate the total energy in joules over 2 hours.**
3. **Efficiency Adjustment:**
- Since the circuit is only 90% efficient, the actual energy input needed is:
\[
\text{Energy Input} = \frac{\text{Energy Output}}{0.9}
\]
4. **Power Calculation:**
- Power (P) is energy divided by time:
\[
P = \frac{\text{Total Energy in joules}}{\text{Time in seconds}}
\]
5. **Voltage Calculation:**
- Using the power and resistance, calculate the input voltage (V) using Ohm's Law:
\[
P = \frac{V^2}{R} \rightarrow V = \sqrt{P \times R}
\]
6. **Conclusion:**
- The input voltage is synthesized to balance energy requirements, resistance, and efficiency constraints for optimal operation in the harsh conditions of space.
---
This explanation breaks down the problem for learners into manageable steps involving energy conversion, efficiency correction, power, and electrical circuit concepts.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd27f30a7-1139-4856-98d7-2b83d1615c97%2F3832c432-25a1-4e95-9932-5446593028a2%2Ft9upx39_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem: Calculating Input Voltage for a Space Heater Circuit**
A small heater is used to keep critical components from getting too cold in space. The device yields an output of 87.5 kilocalories (kcal) of heat energy over a time span of 2 hours. The circuit contains a 0.4 ohm (Ω) resistor and is 90% efficient. What is the input voltage in units of volts (V)?
---
**Solution Explanation:**
1. **Energy Conversion:**
- To find the power requirement in watts (W), convert kilocalories to joules:
\[
\text{1 kcal} = 4184 \text{ joules}
\]
\[
\text{87.5 kcal} = 87.5 \times 4184 \text{ joules}
\]
2. **Calculate the total energy in joules over 2 hours.**
3. **Efficiency Adjustment:**
- Since the circuit is only 90% efficient, the actual energy input needed is:
\[
\text{Energy Input} = \frac{\text{Energy Output}}{0.9}
\]
4. **Power Calculation:**
- Power (P) is energy divided by time:
\[
P = \frac{\text{Total Energy in joules}}{\text{Time in seconds}}
\]
5. **Voltage Calculation:**
- Using the power and resistance, calculate the input voltage (V) using Ohm's Law:
\[
P = \frac{V^2}{R} \rightarrow V = \sqrt{P \times R}
\]
6. **Conclusion:**
- The input voltage is synthesized to balance energy requirements, resistance, and efficiency constraints for optimal operation in the harsh conditions of space.
---
This explanation breaks down the problem for learners into manageable steps involving energy conversion, efficiency correction, power, and electrical circuit concepts.
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