the temperature of a point (x,y) on a unit circle is given by T(x,y)=1+xy. Find the temperature of two hottest points on the circle
Answers
The temperature of the two hottest points on the unit circle is 1.
Given,
Temperature of point (x,y) = 1 + xy.
To find,
Temperature of two hottest points.
Solution,
To find the temperature of the two hottest points on the unit circle, we need to find the maximum value of the temperature function,
T(x,y) = 1 + xy.
Since the unit circle has a radius of 1, we know that x² + y² = 1. We can substitute this equation into the temperature function to get
T(x,y) = 1 + xy = 1 + xy(x² + y²)
T(x,y) = 1 + x³y + xy³.
To find the maximum value of this function, we can take the partial derivative of T(x,y) with respect to x and y, and set both equal to 0. This gives us the following equations:
3x²y + y³ = 0
x³ + 3xy² = 0
Solving these equations simultaneously, we find that x = y = 0 is a solution. This corresponds to the point (0,0) on the unit circle, which is not on the circle.
To find the actual maximum values of T(x,y) on the unit circle, we can use the fact that the unit circle is symmetric about the x- and y-axis.
This means that the maximum values of T(x,y) will occur at the points
(1,0) and (0,1), which are the two hottest points on the unit circle.
The temperature at these points is T(1,0) = 1 + 1 ₓ 0 = 1 and T(0,1) = 1 + 0 ₓ 1 = 1.
Thus, the temperature of the two hottest points on the unit circle is 1.
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Answer:-The temperature of the two hottest points on the unit circle is 1.
Which is at (1,0) and (0,1).
Stepwise Explanation:-
the temperature of a point (x,y) on a unit circle is given by T(x,y)=1+xy. Find the temperature of two hottest points on the circle
Given,
Temperature of point (x,y) = 1 + xy.
To find,
Temperature of two hottest points.
Solution,
To find the temperature of the two hottest points on the unit circle, we need to find the maximum value of the temperature function,
T(x,y) = 1 + xy.
Since the unit circle has a radius of 1, we know that x² + y² = 1. We can substitute this equation into the temperature function to get
T(x,y) = 1 + xy = 1 + xy(x² + y²)
T(x,y) = 1 + x³y + xy³.
To find the maximum value of this function, we can take the partial derivative of T(x,y) with respect to x and y, and set both equal to 0. This gives us the following equations:
3x²y + y³ = 0
x³ + 3xy² = 0
Solving these equations simultaneously, we find that x = y = 0 is a solution. This corresponds to the point (0,0) on the unit circle, which is not on the circle.
To find the actual maximum values of T(x,y) on the unit circle, we can use the fact that the unit circle is symmetric about the x- and y-axis.
This means that the maximum values of T(x,y) will occur at the points
(1,0) and (0,1), which are the two hottest points on the unit circle.
The temperature at these points is T(1,0) = 1 + 1 ₓ 0 = 1 and T(0,1) = 1 + 0 ₓ 1 = 1.
Thus, the temperature of the two hottest points on the unit circle is 1.
Which is at (1,0) and (0,1).
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