Current Flow In electric current flow, it is found that the resistance offered by a fixed length of wire of a given material varies inversely as the square of the diameter of the wire. If a wire 0.01 in. in diameter has a resistance of 0.4 ohm, what is the resistance of a wire of the same length and material with diameter 0.03 in., to the nearest ten-thousandth of an ohm?

Verified step by step guidance

1. Identify the relationship given: resistance \( R \) varies inversely as the square of the diameter \( d \). This means we can write the equation as \( R = \frac{k}{d^2} \), where \( k \) is a constant.
2. Use the given values to find the constant \( k \). Substitute \( R = 0.4 \) ohms and \( d = 0.01 \) inches into the equation: \( 0.4 = \frac{k}{(0.01)^2} \).
3. Solve for \( k \) by multiplying both sides of the equation by \( (0.01)^2 \): \( k = 0.4 \times (0.01)^2 \).
4. Now, use the constant \( k \) to find the resistance \( R \) when the diameter \( d = 0.03 \) inches. Substitute into the formula: \( R = \frac{k}{(0.03)^2} \).
5. Calculate the value of \( R \) from the above expression and round it to the nearest ten-thousandth of an ohm.

Key Concepts

Inverse Variation

Inverse variation describes a relationship where one quantity increases as another decreases, such that their product is constant. In this problem, resistance varies inversely as the square of the diameter, meaning resistance × (diameter)² = constant. Understanding this helps set up the equation to find unknown resistance.

Setting Up and Solving Proportions

Proportions express equality between two ratios or fractions. Here, since resistance varies inversely with the square of diameter, you can set up a proportion comparing the known resistance and diameter squared to the unknown resistance and its diameter squared. Solving this proportion yields the desired resistance.

Rounding to a Specified Decimal Place

Rounding is the process of limiting a number to a certain number of decimal places for precision. The problem asks for the resistance rounded to the nearest ten-thousandth (four decimal places). Proper rounding ensures the final answer is both accurate and appropriately precise.