Problem 55 - 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 maintaining the voltage constant, the resistance should be reduced to one third of its original value.

Step 1: Identify The Variables From The Problem

From the exercise, we know that heat generated (H) varies directly with the square of the voltage (V^2) and inversely with the resistance (R). The formula representing this relationship is \( H = k \cdot \frac{V^2}{R} \).

Step 2: Adjust The Formula For Tripling The Heat

Since the task is to triple the heat generated while keeping the voltage constant, the equation becomes \( 3H = k \cdot \frac{V^2}{R_new} \). Note that R_new is the new resistance required to triple the heat

Step 3: Solve For The New Resistance

Next, isolate R_new in the equation. Hereto, we can equate the two relationships for H from Step 1 and Step 2, as the constant k and the voltage V remain unchanged. So \( k \cdot \frac{V^2}{R} = 3 \cdot k \cdot \frac{V^2}{R_new} \). This simplifies to \( \frac{R_new}{R} = \frac{1}{3} \). Therefore, to triple the heat generated by the stove element, the resistance should be reduced to one third of its original value.