Question

Transpose the formula $$Q=\lambda A\left(\frac{T_2-T_1}{\ell }\right)$$ to make $\ell$ the subject.

Short answer

Question: Given the formula for heat transfer rate, Q = λA(T2 - T1)/ℓ, transpose the formula to make ℓ the subject. Answer: The transposed formula is ℓ = (T2 - T1)λA/Q.

Step-by-step solution

Divide both sides by λA

To isolate ℓ, we first need to get rid of the constants λ and A. We do this by dividing both sides of the equation by λA: $$\frac{Q}{\lambda A}=\left(\frac{T_2-T_1}{\ell }\right)$$

Swap sides of the equation

For easier understanding, let's move the ℓ-containing side closer to the beginning of the equation: $$\frac{T_2-T_1}{\ell }=\frac{Q}{\lambda A}$$

Multiply both sides by ℓ

We want to completely isolate ℓ, so let's multiply both sides of the equation by ℓ to remove the ℓ from the denominator: $$T_2-T_1=\frac{Q}{\lambda A}\cdot \ell$$

Divide both sides by (Q/λA)

Finally, divide both sides by the fraction (Q/λA) to get the expression for ℓ: $$\ell =\frac{T_2-T_1}{\frac{Q}{\lambda A}}$$

Simplify the expression

To simplify the expression for ℓ, we can multiply both top and bottom by λA in the fraction to get rid of the fraction within the fraction: $$\ell =\frac{(T_2-T_1)\cdot \lambda A}{Q}$$ Now the formula is transposed, and ℓ is the subject.

Key concepts

Algebraic Manipulation

Algebraic manipulation is an essential skill in mathematics that allows us to modify equations to suit our needs. In the context of transposing formulas, this skill helps in rearranging the components of an equation to solve for a specific variable.

In the given exercise, algebraic manipulation is used to make $\ell$ the subject of the heat transfer equation. This process involves several steps like dividing both sides by constant values, swapping sides, and multiplying or dividing to eliminate fractions. Mastery of these techniques enables students to solve for any variable in a formula, making it a versatile tool for tackling various mathematical challenges.

Short and precise actions are used in algebraic manipulation to create a clear pathway towards isolating the variable of interest. Remember, each step should bring you closer to having the desired variable on one side of the equation and all other terms on the opposite side.

Isolating Variables

Isolating variables is a foundational concept in algebra that involves rearranging an equation so that a single variable stands alone on one side of the equality. This is crucial when solving equations or transposing formulas, like in our exercise for re-writing the heat transfer equation.

To isolate a variable, one must undo the operations surrounding it through reverse operations. If a variable is multiplied by a number, we divide both sides by that number, and vice versa. It's a bit like untying knots: each action is aimed at freeing the variable from its constraints in the equation.

Strategic use of the order of operations and understanding how to manipulate fractions are key when isolating variables. This ensures that we can arrive at the simplest form of the equation necessary to solve for our variable of interest clearly and effectively.

Mathematical Thermodynamics

Mathematical thermodynamics is a branch of physics that applies mathematical methods to study the interrelation between heat, work, temperature, and energy. The equations used in this field, like the heat transfer equation in our exercise, describe how thermal energy is transferred within systems and how it affects physical properties.

Understanding the variables in thermodynamic equations, such as temperature difference ($T_2-T_1$), heat transfer rate ($Q$), material properties ($\lambda$), and distance ($\ell$), is crucial. This knowledge not only assists in correctly rearranging formulas but also in comprehending the physical phenomena they represent. It's by mastering the mathematical side of thermodynamics that students can better appreciate the science behind the energy exchanges happening all around us.

Heat Transfer Equation

The heat transfer equation is a scientific expression that quantifies the flow of thermal energy, or heat, from one body or system to another as a result of a temperature difference. In our exercise, the formula $Q=\lambda A\left(\frac{T_2-T_1}{\ell }\right)$ serves this exact purpose.

In this equation, $Q$ represents the heat transfer rate, $\lambda$ is the material's thermal conductivity, $A$ is the area through which heat is transferred, $T_1$ and $T_2$ are the initial and final temperatures, and $\ell$ is the thickness of the material through which heat is being transferred.

Each element of this equation has a specific role in defining how heat is conducted, with higher thermal conductivity or greater temperature differences resulting in increased heat flow. By understanding and correctly manipulating the heat transfer equation, students can predict and calculate thermal responses in a variety of practical applications, from engineering to environmental science.