Solve each problem. Current FlowIn 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 for the wire with diameter \$0.03\( inches by substituting into the formula: \)R = \frac{k}{(0.03)^2}$.
5. Calculate the value of \(R\) using the expression from step 4, and round the result 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.

Formulating and Solving Equations

To solve the problem, set up an equation using the given values and the inverse variation relationship. Substitute known values to find the constant, then use it to calculate the unknown resistance for the new diameter.

Precision and Rounding

After calculating the resistance, round the result to the specified precision—in this case, to the nearest ten-thousandth of an ohm. Proper rounding ensures the answer meets the problem's accuracy requirements.