Problem 57 - Calculus Volume 3

Question

A thin plate made of iron is located in the \(x y\) -plane. The temperature \(T\) in degrees Celsius at a point \(P(x, y)\) is inversely proportional to the square of its distance from the origin. Express \(T\) as a function of \(x\) and \(y\).

Step-by-Step Solution

Answer

The temperature function is \( T(x, y) = \frac{k}{x^2 + y^2} \).

Step 1: Understand Inverse Proportionality

The problem states that the temperature \( T \) is inversely proportional to the square of the distance from the origin. If \( T \) is inversely proportional to \( d^2 \), where \( d \) is the distance from the origin, then\[ T \propto \frac{1}{d^2} \]This implies that\[ T = \frac{k}{d^2} \]for some constant \( k \).

Step 2: Calculate Distance from Origin

The distance \( d \) from the origin \((0, 0)\) to a point \( P(x, y) \) in the \(xy\)-plane can be calculated using the distance formula:\[ d = \sqrt{x^2 + y^2} \]

Step 3: Substitute Distance into Proportionality

Since \( T = \frac{k}{d^2} \) and \( d = \sqrt{x^2 + y^2} \), substitute \( d \) in the equation:\[ T = \frac{k}{(\sqrt{x^2 + y^2})^2} \]This simplifies to:\[ T = \frac{k}{x^2 + y^2} \]

Step 4: Express Temperature as a Function

Now, express \( T \) explicitly as a function of \( x \) and \( y \):\[ T(x, y) = \frac{k}{x^2 + y^2} \]Where \( k \) is a constant that depends on the specific conditions of the problem.

Key Concepts

Inverse Proportionality

In mathematics, when we say that a quantity is inversely proportional to another, it means as one quantity increases, the other decreases, and vice versa. Imagine you have two variables, \( T \) and \( d \), and they are related such that \( T \) is inversely proportional to \( d^2 \). This can be written as \( T \propto \frac{1}{d^2} \). To translate this into an equation, we introduce a constant \( k \), leading to the equation \( T = \frac{k}{d^2} \). Here, \( k \) is a positive constant that represents a specific condition or factor influencing the relationship. In our exercise scenario, \( k \) might relate to specifics about the material or environment, such as heat conductivity or external conditions.

Understanding inverse proportionality can help you identify how changes in one value affect another. This concept is foundational in physics and engineering, where understanding the relationships between different quantities is essential.

Distance Formula

The distance formula helps us calculate the distance between two points in a plane. It originates from the Pythagorean Theorem, which is used to find the length of the hypotenuse of a right triangle. When determining the distance \( d \) from the origin \((0, 0)\) to a point \( P(x, y) \) in the Cartesian plane, we use the formula:

\[d = \sqrt{x^2 + y^2} \]
The formula essentially calculates the straight-line distance as one might use a ruler. In our case, this straight-line distance is from the origin, which makes it easier to apply formulas that depend on radial symmetry, such as those encountered with circular or spherical objects.

• The \( x^2 + y^2 \) under the square root helps us to compute the squared distance.
• This expression considers the horizontal and vertical shifts needed to reach \( P(x, y) \) from the origin.

Knowing how to apply the distance formula is vital in multivariable calculus because it translates physical scenarios into mathematical models by providing a precise measure of spatial separation.

Temperature Distribution

Temperature distribution in a physical space, like a metal plate, often requires a mathematical model to understand how temperature varies across different points. In our exercise, the temperature \( T \) at any point \( P(x, y) \) is given by the expression:

\[T(x, y) = \frac{k}{x^2 + y^2} \]
This means that the temperature at any point \( (x,y) \) depends on its distance from the origin. The formula captures how the temperature decreases as the distance from the origin increases:

• The temperature value at each point is calculated inversely with the square of the distance.
• As \( (x^2 + y^2) \) increases, indicating increased distance, \( T \) becomes smaller.

Temperature distribution modeling is crucial in many engineering disciplines, where understanding the heat flow and the spatial variation of temperatures is important in design and analysis. This exercise illustrates how such a mathematical model functions, providing insight into physical behaviors.