Problem 32
Question
The temperature at a point \((x, y)\) of a flat metal plate is \(t\) degrees and \(t=4 x^{2}+2 y^{2} .\) Draw the isothermals for \(t=12\), \(8,4,1\), and \(0 .\)
Step-by-Step Solution
Verified Answer
The isothermals for t = 12, 8, 4, 1, and 0 are ellipses with the semi-axes derived accordingly: a^2=6, 4, 2, 0.5, 0 and b^2=3, 2, 1, 0.25, 0 respectively.
1Step 1 - Set up Isothermal Equations
Isothermals are lines where the temperature is constant. For each given temperature, set up the equation for the isothermal by substituting the value of t into the given temperature formula. The formula given is: t = 4x^2 + 2y^2.Start with each given temperature value: t = 12, t = 8, t = 4, t = 1, and t = 0.
2Step 2 - Simplify Equations for Each Temperature
Substitute each temperature value one by one and simplify: For t = 12: 4x^2 + 2y^2 = 12 For t = 8: 4x^2 + 2y^2 = 8 For t = 4: 4x^2 + 2y^2 = 4 For t = 1: 4x^2 + 2y^2 = 1 For t = 0: 4x^2 + 2y^2 = 0
3Step 3 - Factor and Normalize Equations
Simplify each equation:For t = 12: 2x^2 + y^2 = 6 For t = 8: 2x^2 + y^2 = 4 For t = 4: 2x^2 + y^2 = 2 For t = 1: 2x^2 + y^2 = 0.5 For t = 0: x^2 = 0 and y^2 = 0
4Step 4 - Identify Isothermals
Recognize that these are equations of ellipses centered at the origin. Each equation describes an isothermal curve. Plot these ellipses as follows: For t = 12: Ellipse with a^2 = 6 (horizontal semi-axis) and b^2 = 3 (vertical semi-axis) For t = 8: Ellipse with a^2 = 4 (horizontal semi-axis) and b^2 = 2 (vertical semi-axis) For t = 4: Ellipse with a^2 = 2 (horizontal semi-axis) and b^2 = 1 (vertical semi-axis) For t = 1: Ellipse with a^2 = 0.5 (horizontal semi-axis) and b^2 = 0.25 (vertical semi-axis) For t = 0: Point at origin (0,0)
Key Concepts
Temperature DistributionEllipsesAnalytic GeometryMetal Plate
Temperature Distribution
In this exercise, we need to understand how the temperature is distributed across a flat metal plate.
The temperature at any point on the plate is given by the formula: \(t = 4x^2 + 2y^2\).
This means that the temperature depends on both the x and y coordinates of the point.
The value of \(t\) changes as we move to different points on the plate.
Areas where the temperature is constant are called isothermals.
Drawing these isothermals helps us visualize how temperature varies across the plate.
By substituting different values of \(t\) into the formula, we can find equations defining these isothermals.
The temperature at any point on the plate is given by the formula: \(t = 4x^2 + 2y^2\).
This means that the temperature depends on both the x and y coordinates of the point.
The value of \(t\) changes as we move to different points on the plate.
Areas where the temperature is constant are called isothermals.
Drawing these isothermals helps us visualize how temperature varies across the plate.
By substituting different values of \(t\) into the formula, we can find equations defining these isothermals.
Ellipses
Ellipses play a crucial role in understanding the isothermals of this temperature distribution.
An ellipse is a geometric shape that is elongated in one direction.
The general equation for an ellipse centered at the origin is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).
In this exercise, simplifying the temperature equation for different values of \( t \) gives us different ellipses.
For example, for \( t = 12 \), we get the ellipse equation \( 2x^2 + y^2 = 6 \).
This can be rewritten to fit the general ellipse form: \( \frac{x^2}{3} + \frac{y^2}{6} = 1 \).
Here, \( a^2 \) and \( b^2 \) are the squares of the lengths of the ellipse's semi-axes along the x and y directions.
An ellipse is a geometric shape that is elongated in one direction.
The general equation for an ellipse centered at the origin is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).
In this exercise, simplifying the temperature equation for different values of \( t \) gives us different ellipses.
For example, for \( t = 12 \), we get the ellipse equation \( 2x^2 + y^2 = 6 \).
This can be rewritten to fit the general ellipse form: \( \frac{x^2}{3} + \frac{y^2}{6} = 1 \).
Here, \( a^2 \) and \( b^2 \) are the squares of the lengths of the ellipse's semi-axes along the x and y directions.
Analytic Geometry
Analytic geometry allows us to study geometric shapes using algebra and equations.
For our problem, we use the equation of the temperature distribution to find geometric shapes (ellipses).
Each shape corresponds to a specific temperature (isothermal).
By analyzing these equations, we can graph the ellipses on a coordinate plane.
This visual representation of the ellipses helps us understand the temperature variation.
Using the equations, we can accurately plot these shapes and see how the different temperatures form different-sized ellipses.
Each ellipse's size is directly related to the temperature value.
For our problem, we use the equation of the temperature distribution to find geometric shapes (ellipses).
Each shape corresponds to a specific temperature (isothermal).
By analyzing these equations, we can graph the ellipses on a coordinate plane.
This visual representation of the ellipses helps us understand the temperature variation.
Using the equations, we can accurately plot these shapes and see how the different temperatures form different-sized ellipses.
Each ellipse's size is directly related to the temperature value.
Metal Plate
The context of this exercise is a flat metal plate.
Metal plates often have uniform properties, making them an ideal surface for studying temperature distribution.
Temperature distribution in such a plate can be influenced by various factors like heat sources or environmental conditions.
However, in this exercise, we assume a mathematical model where temperature is a function of position.
This simplification allows us to use the given formula and focus on understanding the isothermal curves.
Knowing how temperature varies across the metal plate helps in numerous practical applications such as material design and thermal management.
Metal plates often have uniform properties, making them an ideal surface for studying temperature distribution.
Temperature distribution in such a plate can be influenced by various factors like heat sources or environmental conditions.
However, in this exercise, we assume a mathematical model where temperature is a function of position.
This simplification allows us to use the given formula and focus on understanding the isothermal curves.
Knowing how temperature varies across the metal plate helps in numerous practical applications such as material design and thermal management.
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