Question

the temperature at the point (x, y) on a metal plate is T(x, y) = 64x² - 64xy + 16y². An ant on the plate walks around the circle of radius 5 centered at the origin. What are the highest and lowest temperatures encountered by the ant?

Answer 1

To find the highest and lowest temperatures an ant encounters while walking around a circle on a metal plate, we use the temperature function T(x, y) and the circle's equation to find critical points by differentiation and then evaluate T(x, y) at these points.

Step-by-step explanation:

The temperature at the point (x, y) on a metal plate is given by the equation T(x, y) = 64x² - 64xy + 16y². If an ant walks around a circle of radius 5 centered at the origin, we can use the equation of the circle, which is x² + y² = 25, to find the highest and lowest temperatures encountered by the ant on this path.

To find the extreme temperatures while the ant stays on the circle, we can substitute y with √(25 - x²) or -√(25 - x²) into the temperature function and then differentiate with respect to x to find the critical points that will give us the maximum and minimum temperatures. We would use the fact that for a point (x, y) on the circle, y can be expressed either as positive or negative the square root of (25 - x²) since it lies on a circle.

Step-by-Step Analysis:

1. Substitute y in the equation T(x, y) using the circle's equation.
2. Differentiate the resulting equation with respect to x to find the derivative.
3. Set the derivative equal to zero and solve for x to find the critical points.
4. Evaluate T(x, y) at these critical points to find the temperatures.
5. Select the highest and lowest values obtained as the maximum and minimum temperatures.

With respect to the temperature mentioned in other contexts, such as the one related to absolute zero or the specific heat of a substance, these are separate topics that explain how temperature behaves under different conditions but aren't directly involved in solving this problem.