Problem 99 - Intermediate Algebra

Question

Solve each problem by writing a variation model. Electronics. The resistance of a wire is directly proportional to the length of the wire and inversely proportional to the square of the diameter of the wire. If the resistance is 11.2 ohms in a 80 -foot-long wire with diameter 0.01 inch, what is the resistance in a 160 -foot-long wire with diameter 0.04 inch?

Step-by-Step Solution

Answer

The resistance is 1.4 ohms.

Step 1: Understand the Proportional Relationship

We know that the resistance \( R \) is directly proportional to the length \( L \) of the wire and inversely proportional to the square of the diameter \( d \) of the wire. This can be expressed by the relationship: \( R = k \cdot \frac{L}{d^2} \) where \( k \) is a constant of proportionality.

Step 2: Find the Constant of Proportionality, k

Given that \( R = 11.2 \) ohms when \( L = 80 \) feet and \( d = 0.01 \) inches, we substitute these values into the equation: \( 11.2 = k \cdot \frac{80}{(0.01)^2} \). We can solve for \( k \) by rearranging the equation, giving us \( k = \frac{11.2 \times 0.0001}{80} = 0.000014 \).

Step 3: Calculate New Resistance with Found k

Now, use the value of \( k \) to calculate the resistance when the wire's length and diameter change. For a wire of length \( 160 \) and diameter \( 0.04 \), the formula becomes: \( R = 0.000014 \cdot \frac{160}{(0.04)^2} \).

Step 4: Solve for New Resistance

Substitute the values into the equation from Step 3: \( R = 0.000014 \cdot \frac{160}{0.0016} \). Simplify to find \( R = 0.000014 \cdot 100000 = 1.4 \).

Step 5: Verify and Conclude

After solving the equation, verify by ensuring all calculations, particularly the arithmetic, align with the defined mathematical relationship. The resistance of the 160-foot-long wire with a 0.04-inch diameter is 1.4 ohms.

Key Concepts

Variation Model

A variation model is a mathematical way to express how one quantity changes when another quantity changes. It's like a formula that shows relationships between different variables. In our exercise, we're dealing with the resistance of a wire. Resistance depends on two factors:

• The length of the wire.
• The square of the diameter of the wire.

The variation model we use here is a combination of direct and inverse proportionality, represented as:\[ R = k \cdot \frac{L}{d^2} \]Here, \( R \) is the resistance, \( L \) is the length, \( d \) is the diameter, and \( k \) is a constant. This model helps us solve problems by relating these variables and showing how changing one affects the others. It's crucial to find the constant \( k \) first to use the model effectively. Once \( k \) is known, predictions and calculations become easier to manage.

Resistance Calculation

Calculating resistance involves a clear understanding of the connection between resistance, length, and diameter. First, identify the known values from the problem. In this case:

• Length \( L = 80 \) feet when resistance \( R = 11.2 \) ohms.
• Diameter \( d = 0.01 \) inches.

To find the constant of proportionality \( k \), substitute these values into the model:\[ 11.2 = k \cdot \frac{80}{(0.01)^2} \]This equation lets us solve for \( k \), which then allows us to calculate the resistance for other wire lengths and diameters. For a new scenario where:

• Length \( L = 160 \) feet.
• Diameter \( d = 0.04 \) inches.

We use the same model:\[ R = k \cdot \frac{160}{(0.04)^2} \]This step illustrates how known constants help compute resistance in various conditions by just plugging in different lengths and diameters.

Inverse Proportionality

Inverse proportionality is a concept where one quantity increases as another decreases. For instance, in our exercise, resistance \( R \) is inversely proportional to the square of diameter \( d \). What this means is:

• If the diameter increases, the resistance decreases, assuming length remains the same.
• If the diameter decreases, the resistance increases.

This contrasts with direct proportionality where both quantities move in the same direction. The formula for this inverse relationship in the context of resistance is:\[ R = k \cdot \frac{L}{d^2} \]By understanding this inverse relationship, you can predict how changes in diameter affect resistance. It's essential for problem-solving in electronics, where precisely managing these relationships is critical for designing and troubleshooting circuits.