Problem 111 - College Algebra with Modeling and Visualization

Question

The electrical resistance \(R\) of a wire varies inversely as the square of its diameter \(d\). If a 25-foot wire with a diameter of 2 millimeters has a resistance of 0.5 ohm, find the resistance of a wire having the same length and a diameter of 3 millimeters.

Step-by-Step Solution

Answer

The resistance is approximately 0.222 ohms.

Step 1: Understand the relationship

The resistance \( R \) of the wire varies inversely as the square of its diameter \( d \). Mathematically, this can be expressed as:\[R = \frac{k}{d^2}\]where \( k \) is a constant of proportionality.

Step 2: Find constant \( k \)

We use the information provided for the wire with diameter 2 millimeters.For that wire:\[d = 2, \ R = 0.5 \]Substitute these values into the equation:\[0.5 = \frac{k}{2^2}\]\[0.5 = \frac{k}{4}\]Multiply both sides by 4 to solve for \( k \):\[k = 0.5 \times 4 = 2\]

Step 3: Use the constant to find new resistance

Now, find the resistance of a wire with diameter 3 millimeters using the constant \( k = 2 \).Substitute \( d = 3 \) and \( k = 2 \) back into the equation:\[R = \frac{2}{3^2}\]\[R = \frac{2}{9}\]Calculate \( \frac{2}{9} \), which is approximately \( 0.222 \, \text{ohms} \).

Key Concepts

Electrical Resistance

Electrical resistance is a fundamental concept when it comes to understanding how well a wire can conduct electricity. Essentially, resistance measures the degree to which a material opposes the flow of electric current. The unit for measuring electrical resistance is the ohm (Ω). When we talk about a wire's resistance, we consider several factors, including material, length, and diameter.

In this exercise, we focus particularly on the inverse relationship with the diameter. A wire's resistance changes depending on its diameter. With a smaller diameter, the resistance is higher, making it harder for electricity to flow. Conversely, a larger diameter means lower resistance, allowing for easier electricity flow.

Keep in mind that when we say the resistance varies "inversely" with the diameter's square, this means as the diameter increases, resistance decreases according to the formula given: \[R = \frac{k}{d^2}\] Understanding this inverse variation is crucial because electrical engineers use this principle to design circuits that are efficient and safe.

Wire Diameter

The diameter of the wire plays a critical role in determining its resistance. Diameter is simply the thickness of the wire, usually measured in millimeters for precision tasks.

If you imagine a wire as a cylindrical structure, the diameter would be a straight line passing through the center and touching the edges. Larger diameters have more area for electrons to travel through, thus experiencing less resistance.

In practical terms, if you're designing a device or working on a project that involves sending electricity over a distance, you will decide on wire diameter based on how much resistance you are willing to accommodate. In this exercise, you see how the diameter dramatically affects resistance and learn to calculate that change with precision.

Constant of Proportionality

The constant of proportionality (\(k\)) plays a vital role in mathematical equations that describe relationships between different physical quantities. In this instance, it's used in the formula \[R = \frac{k}{d^2}\] where it links resistance and the square of the wire's diameter. This constant allows us to quantify how these variables are related.

We established the value of \(k\) by using known values from our exercise: with a 2-millimeter diameter giving a resistance of 0.5 ohms, we found \(k\) to be 2 by rearranging the formula: \(0.5 = \frac{k}{2^2}\).

Once determined, this constant can be reused to calculate resistance for different diameters of the same wire material. It's a significant tool because it aids in predicting how changes in physical dimensions can affect electrical properties. This is especially useful in engineering and electronics design to ensure optimal performance.