Problem 55 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. 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 with the voltage remaining constant, the resistance needs to be reduced to one third of its original value.

Step 1: Understand the Statement and Relation

We start by identifying the direct and inverse variations present in the statement. Here, the statement says the heat (H) generated by a stove element varies directly as the square of the voltage (V) and inversely as the resistance (R). This can be written in mathematical form as \( H = k*V^2/R \), where k is the constant of variation.

Step 2: Determine the Effect of Tripling the Heat

We are asked to find out how to triple the amount of heat. This would mean that \( H \) in the equation now becomes \( 3H \). Thus, the equation becomes \( 3H = k*V^2/R \). According to the problem, the voltage doesn't change, so \( V^2 \) remains the same in the equation.

Step 3: Solving for New Resistance Value

We solve for the new value of \( R \) which gives \( R = k*V^2/(3H) \). From the original equation, we know that \( R = k*V^2/H \), so dividing both sides of the equations by 3 we get the new value of \( R = R/3 \). This means that to triple the amount of heat generated, we need to reduce the resistance to a third of its original value.

Key Concepts

Voltage

When we talk about voltage, we refer to the electric potential difference between two points. It's like the force that pushes electric charges through a conductor, such as a wire. Voltage is measured in volts (V). It's crucial because it determines how much electricity flows through a circuit.
In our exercise situation, electricity drives the stove element to produce heat. So, when thinking of voltage, imagine it as the pressure pushing water through a hose. If the pressure is high, more water is pushed through rapidly. Similarly, higher voltage means more electrical energy flows through the element, increasing the potential to produce heat.

• Voltage is the "pressure" pushing electric current.
• Measured in volts (V).
• Higher voltage generally means more energy flows.

Remember, in our given problem, we assume that the voltage stays constant. This means we're not changing this electric pressure factor when trying to adjust the heat output.

Resistance

Resistance is another critical concept in electricity, referring to how much an object opposes the flow of electric current. It is measured in ohms (Ω). Imagine resistance as the friction that works against the electric current—like how rocks in a stream slow the flow of water.
Resistance in a stove element is crucial because it transforms electrical energy into heat. The equation we're dealing with shows that resistance has an inverse effect on heat generation. This means:

• Higher resistance results in less heat produced.
• Lowering resistance increases the heat output.

Therefore, to generate more heat, especially when voltage is constant, we must decrease the resistance. In our situation, to triple the heat, the resistance must be reduced to a third of its original amount. Think of it as clearing some rocks from the stream. With fewer obstacles, more water, or in this case, more electric current, can flow through, producing more heat.

Heat Generation

Heat generation in an electrical device is the transformation of electrical energy into heat energy. This occurs due to the resistance in the electrical path, such as the wiring in a stove element.
In our context, understanding how heat is generated is essential. The heat produced is directly related to the voltage squared and inversely related to the resistance. This relationship is crucial for controlling the heat output. Imagine baking where the temperature needs to be just right.
Points to remember:

• Heat increases with higher voltage and lower resistance.
• Efficiently managing resistance allows for optimal heat output.

For instance, if we want to bake at a higher temperature without changing the oven's settings, we need to make adjustments such as lowering the resistance, thus boosting the heat generated. The principle here serves as a fundamental guide to manipulating heat in various electrical applications.

Mathematical Model

A mathematical model helps us represent real-world scenarios with equations. In the case of our stove element, the mathematical model given is: \[ H = \frac{k \, \cdot \, V^2}{R} \]where:

• \(H\) stands for heat generated,
• \(V\) is the voltage,
• \(R\) is the resistance, and
• \(k\) is a constant.

This equation incorporates both direct and inverse variations. Direct variation indicates that if one factor increases, the output (heat, here) also increases – as in the case of voltage. Inverse variation shows that as the resistance factor increases, the output decreases.

By manipulating this equation, we can understand how to alter factors to get the desired heat output. For instance, to triple the heat while keeping other factors constant, the model tells us to reduce the resistance to one-third. Such models are powerful as they provide guidelines and foresight in engineering and design, simplifying complex physical relationships into manageable mathematical expressions.