Solve each problem. Resistance of a WireThe resistance in ohms of a platinum wire temperature sensor varies directly as the temperature in kelvins (K). If the resistance is 646 ohms at a temperature of 190 K, find the resistance at a temperature of 250 K.

Verified step by step guidance

1. Identify the relationship: The resistance $R$ varies directly with the temperature $T$, which can be expressed as $R=kT$, where $k$ is the constant of proportionality.
2. Use the given values to find $k$: Substitute $R=646$ ohms and $T=190$ K into the equation $R=kT$ to get $646=k\times 190$.
3. Solve for $k$: Rearrange the equation to find $k=\frac{646}{190}$.
4. Use the value of $k$ to find the resistance at 250 K: Substitute $k$ and $T=250$ K into the equation $R=kT$ to get $R=k\times 250$.
5. Calculate the resistance: Multiply the value of $k$ by 250 to find the resistance at 250 K.

Key Concepts

Direct Variation

Direct variation describes a relationship between two variables where one variable is a constant multiple of the other. In this case, the resistance of the wire varies directly with temperature, meaning that as the temperature increases, the resistance increases proportionally. This relationship can be expressed mathematically as R = kT, where R is resistance, T is temperature, and k is a constant.

Proportionality Constant

The proportionality constant, often denoted as 'k', is the factor that relates the two directly varying quantities. It can be determined using known values from the relationship. For example, if the resistance is 646 ohms at 190 K, the constant can be calculated as k = R/T, which allows us to find resistance at other temperatures by rearranging the direct variation formula.

Linear Equations

Linear equations represent relationships that can be graphed as straight lines. In the context of direct variation, the equation R = kT is linear, where R is plotted on the y-axis and T on the x-axis. Understanding how to manipulate and solve linear equations is essential for finding unknown values, such as resistance at different temperatures in this problem.