Question

Suppose that the temperature at a point (x, y) on a metal plate is \( T(x, y) = 4x^2 - 4xy + y^2 \). An ant, walking on the plate, traverses a circle of radius 5 centered at the origin. What is the highest temperature encountered by the ant?

• A. 125
• B. 120
• C. 0
• D. 25

Hint:

First, simplify the expression if possible. Here, \( 4x^2 - 4xy + y^2 \) simplifies to \( (2x - y)^2 \). To find the maximum of a linear expression like \( ax+by \) on a circle \( x^2+y^2=r^2 \), the maximum value is \( r\sqrt{a^2+b^2} \). Here, we need to maximize \( (2x-y)^2 \), so its maximum is \( (5\sqrt{2^2+(-1)^2})^2 = (5\sqrt{5})^2 = 125 \).

Answer 1

The Correct Option is A

The temperature function is \( T(x, y) = 4x^2 - 4xy + y^2 \). We can recognize this as a perfect square: \( T(x, y) = (2x - y)^2 \).
The ant is walking on a circle of radius 5 centered at the origin, so the path is described by the equation \( x^2 + y^2 = 5^2 = 25 \).
We need to find the maximum value of \( T(x, y) \) subject to the constraint \( x^2 + y^2 = 25 \).
This means we need to maximize the value of \( (2x - y)^2 \), which is equivalent to maximizing the absolute value of \( |2x - y| \).
Let \( f(x, y) = 2x - y \). We can use the method of Lagrange multipliers or a simpler method using vectors or Cauchy-Schwarz inequality.
Using Cauchy-Schwarz: \[ (2x - 1y)^2 \leq (2^2 + (-1)^2)(x^2 + y^2) \] \[ (2x - y)^2 \leq (4 + 1)(25) \] \[ (2x - y)^2 \leq (5)(25) \] \[ (2x - y)^2 \leq 125 \] The maximum value of \( T(x, y) = (2x - y)^2 \) is therefore 125.