Problem 115 - Elementary and Intermediate Algebra

Question

Blow Dryers. The current \(I\) (in amps), the power \(P\) (in watts), and the resistance \(R\) (in ohms) are related by the formula \(I=\sqrt{\frac{P}{R}} .\) What current is needed for a \(1,200\) -watt hair dryer if the resistance is 16 ohms?

Step-by-Step Solution

Answer

The current needed is approximately 8.66 amps.

Step 1: Identify Given Values

We are given the power \( P = 1200 \) watts and the resistance \( R = 16 \) ohms. We need to find the current \( I \).

Step 2: Write the Formula

We have the formula for current: \( I = \sqrt{\frac{P}{R}} \).

Step 3: Substitute the Given Values

Substitute \( P = 1200 \) and \( R = 16 \) into the formula: \( I = \sqrt{\frac{1200}{16}} \).

Step 4: Calculate the Quotient

First, calculate \( \frac{1200}{16} = 75 \).

Step 5: Find the Square Root

Now, take the square root of 75: \( I = \sqrt{75} \).

Step 6: Simplify the Square Root

The square root can be estimated as \( I \approx 8.66 \) amps.

Key Concepts

Understanding Current Calculation

Ohm's Law is an important principle in physics and electrical engineering that helps us understand how electricity flows through a circuit. When calculating current, we use the specific formula:

• \( I = \sqrt{\frac{P}{R}} \)

This formula allows us to determine the current \( I \) in a circuit based on the known values of power \( P \) and resistance \( R \).
For example, in our exercise, the power of the hair dryer is given as 1200 watts and the resistance is 16 ohms. By substituting these values into the formula, we find the current that is needed. This is done in steps: first calculate the quotient \( \frac{1200}{16} = 75 \), and then find the square root of 75, which gives us the current \( I \approx 8.66 \) amps.
The ability to calculate current accurately is crucial because it tells us the amount of electrical flow, ensuring devices work properly without overload.

Exploring the Power Resistance Formula

The relationship between power, resistance, and current is beautifully captured in Ohm's Law. The power resistance formula is at the core of many electrical circuit calculations. It is expressed as:

• \( I = \sqrt{\frac{P}{R}} \)

This formula derives from the basic equation of electrical power \( P = I^2 \cdot R \), which deciphers how power is related to both current and resistance.
The way this relationship works can be broken down as follows:

• Power \( P \) in watts, indicates the rate at which energy is used.
• Resistance \( R \) in ohms, is a measure of how much a component resists the flow of current.

When you want to find the current for a given power and resistance, you rearrange the power formula, replacing \( I \) with the square root function. Understanding how these elements interact allows engineers and technicians to design better electrical systems by balancing the power and resistance to achieve the needed current.

Insights into Electrical Circuits Analysis

Analyzing electrical circuits involves several core concepts, but understanding how to compute parameters like current, power, and resistance is foundational. With the given Ohm's Law formulas, students are better equipped to deconstruct more complex circuits into understandable components.
When approaching circuit analysis, it's essential to:

• Identify all given parameters such as voltage, resistance, or power.
• Apply the correct formulas for calculations, like the current calculation formula discussed earlier.
• Simplify where possible—use basic algebra to solve for unknowns.

By following these steps, whether dealing with a basic or intricate circuit, students simplify the problem to a manageable level. For example, figuring out the current in a hair dryer by knowing just the power and resistance reflects how simplifying and applying the right formula can make analysis straightforward and clear.
This process is indispensable when dealing with real-world electrical circuits in which multiple components interact, helping one ensure both safety and efficiency.