Problem 61 - College Algebra

Question

The heat generated by a stove element varies directly as the square of the voltage and inversely as the resistance. If the voltage remains constant, what needs to be done to triple the amount of heat generated?

Step-by-Step Solution

Answer

To triple the amount of heat generated while keeping voltage constant, the resistance must be reduced to one-third of its original value.

Step 1: Understand the Problem

The problem statement tells us that the heat generated by a stove element (H) varies directly with the square of the voltage (V^2), and inversely with the resistance (R). This implies that the relationship can be written as a formula: \(H = k \cdot \frac{V^2}{R} \), where k is the constant of proportionality. We need to determine how we can triple the heat generated (increase H to 3H) by adjusting the resistance, while holding the voltage constant.

Step 2: Set up the Equation

To triple the heat generated, we want to find a new value of resistance, say \(R'\), such that \(3H = k \cdot \frac{V^2}{R'}\). This equation represents the new situation where the heat generated is tripled, and resistance has been adjusted accordingly. We also know from the original situation that \(H = k \cdot \frac{V^2}{R}\). Combining these two equations will let us find \(R'\).

Step 3: Solve for New Resistance

Setting the equations \(3H = k \cdot \frac{V^2}{R'}\) and \(H = k \cdot \frac{V^2}{R}\) equal to each other, we get \(3H = H \cdot \frac{R}{R'}\). Dividing both sides by H, we get \(3 = \frac{R}{R'}\). Thus, \(R' = \frac{R}{3}\). To triple the heat generated, the resistance must be reduced to one third of its original value.