On the premise that the square of X divided by 25 plus the square of Y divided by 16 equals 1, the maximum and minimum values of Z equal to x-2y can be obtained

On the premise that the square of X divided by 25 plus the square of Y divided by 16 equals 1, the maximum and minimum values of Z equal to x-2y can be obtained

Given the elliptic equation x ^ 2 / 25 + y ^ 2 / 16 = 1, find the maximum and minimum of Z = x-2y
Solution, method 1: trigonometric substitution, x = 5cosa, y = 4sina, z = x-2y = 5cosa-8sina = √ 89cos (a + b)
So the maximum and minimum are √ 89 and - √ 89
Method 2: linear programming, z = x-2y, that is, y = x / 2 - Z / 2. The linear equation and ellipse should have intersection points. When the drawing is tangent, Z takes the maximum value, simultaneous equations, and the discriminant is equal to 0. In this way, two tangent points can be obtained, and the maximum value and minimum value can be obtained by substituting z = x-2y

Simple calculation: 1155 * (5 / 2 * 3 * 4 + 7 / 3 * 4 * 5 +...) +17/8*9*10+19/9*10*11)