Problem 35 - The Calculus with Analytic Geometry

Question

The temperature at any point \((x, y)\) of a flat plate is \(T\) degrees and \(T=54-\frac{2}{3} x^{2}-4 y^{2} .\) If distance is measured in feet, find the rate of change of the temperature with respect to the distance moved along the plate in the directions of the positive \(x\) and \(y\) axes, respectively, at the point \((3,1)\).

Step-by-Step Solution

Answer

The rate of change of temperature at \( (3,1) \) is -4 in the \(x\) direction and -8 in the \(y\) direction.

Step 1: Understand the Given Formula

The temperature at any point \(x, y\) on the plate is expressed as \(T = 54 - \frac{2}{3} x^2 - 4 y^2\). Our goal is to find the rate of change of temperature with respect to distance moved in the directions of the positive \(x\) and \(y\) axes at the point \( (3,1) \).

Step 2: Find the Partial Derivative with Respect to x

To find the rate of change of temperature in the positive \(x\) direction, calculate the partial derivative of \(T\) with respect to \(x\).\[ \frac{\text{∂}T}{\text{∂}x} = \frac{\text{∂}}{\text{∂}x} \bigg(54 - \frac{2}{3}x^2 - 4y^2 \bigg) = -\frac{4}{3} x \]

Step 3: Evaluate the Partial Derivative with Respect to x at (3,1)

Substitute \(x = 3\) into the partial derivative we found.\[ \frac{\text{∂}T}{\text{∂}x}\bigg|_{(3,1)} = -\frac{4}{3} (3) = -4 \]

Step 4: Find the Partial Derivative with Respect to y

To find the rate of change of temperature in the positive \(y\) direction, calculate the partial derivative of \(T\) with respect to \(y\).\[ \frac{\text{∂}T}{\text{∂}y} = \frac{\text{∂}}{\text{∂}y} \bigg(54 - \frac{2}{3}x^2 - 4y^2 \bigg) = -8y \]

Step 5: Evaluate the Partial Derivative with Respect to y at (3,1)

Substitute \(y = 1\) into the partial derivative we found.\[ \frac{\text{∂}T}{\text{∂}y}\bigg|_{(3,1)} = -8 (1) = -8 \]

Key Concepts

rate of change

In mathematics, the rate of change measures how a quantity varies with respect to another quantity. When dealing with temperature on a flat plate, we might want to know how quickly the temperature changes as we move in certain directions. For example, moving along the positive x or y direction. By computing the partial derivatives of the temperature function, we capture these rates of change.

temperature distribution

In this exercise, the temperature distribution on a flat plate is given by a quadratic equation: \[ T = 54 - \frac{2}{3} x^{2} - 4 y^{2} \]. This equation tells us the temperature at any point \(x, y\) on the plate.

The constants and terms of the equation represent how the temperature varies across the plate. For instance, the term \( - \frac{2}{3} x^2 \) decreases temperature as \( x \) increases, while the term \( - 4 y^2 \) decreases temperature as \( y \) increases.

analytic geometry

Analytic geometry helps us understand geometric shapes using algebra. In this context, it helps us explore the temperature distribution across the plate. By plotting points and curves described by the temperature function, we can see how temperature changes over the plate’s surface.

For example, evaluating the function at multiple points and graphing the results helps us visualize temperature gradients, which are regions where temperature changes rapidly.

partial derivative evaluation

Partial derivatives are essential for finding how a function changes with one variable while keeping others constant. For the given temperature function, we compute partial derivatives with respect to \( x \) and \( y \).

Here’s how:

• Find the partial derivative of \( T \) with respect to \( x \):
  \[ \frac{\partial T}{\partial x} = - \frac{4}{3} x \]


• Evaluate this derivative at the point \( (3,1) \):
  \[ \left. \frac{\partial T}{\partial x} \right|_{(3,1)} = -4 \]


• Find the partial derivative of \( T \) with respect to \( y \):
  \[ \frac{\partial T}{\partial y} = -8 y \]


• Evaluate this derivative at the point \( (3,1) \):
  \[ \left. \frac{\partial T}{\partial y} \right|_{(3,1)} = -8 \]

These steps reveal the temperature changes at \( (3,1) \):
- Changing \( x \) reduces the temperature by 4 degrees per foot.
- Changing \( y \) reduces the temperature by 8 degrees per foot.