The sides of a triangle are x, y, and sqrt(x^2+xy+y^2. Determine the Largest Angle.

clearly √(x^2 + xy + y^2) is greater than either x or y

since the largest angle in any triangle is opposite the largest side .....

in 1990 the life expectancy of males in a certain country was 64.7 years. In 1994 it was 67.3 let E represent the life expectancy in year t and let t represent the number of years since 1990.

What is the linear function E(t) that fits the data? what would the life expectancy of males be in 2009?

To determine the largest angle of a triangle with side lengths given by x, y, and √(x^2 + xy + y^2), we can use the law of cosines.

The law of cosines states that for any triangle with side lengths a, b, and c, and angles A, B, and C opposite to the sides a, b, and c respectively, the following equation holds true:

c^2 = a^2 + b^2 - 2ab * cos(C)

In our case, let's denote the side lengths of the triangle as follows:
a = x
b = y
c = √(x^2 + xy + y^2)

Using the law of cosines, we can rewrite the equation as:
c^2 = x^2 + y^2 - 2xy * cos(C)

Rearranging the equation to solve for cos(C), we get:
cos(C) = (x^2 + y^2 - c^2) / 2xy

To determine the largest angle, we need to find the value of cos(C) that is the smallest (since a larger value of cos(C) corresponds to a smaller angle). Therefore, we need to find the smallest possible value of (x^2 + y^2 - c^2) / 2xy.

To determine this, we can take the derivative of the expression (x^2 + y^2 - c^2) / 2xy with respect to x and y, set them equal to 0, and solve for x and y. This will give us the critical points. However, this approach involves more complex calculations and is beyond the scope of this explanation.

Alternatively, you can use the concept of calculus to find the minimum value of cos(C) by considering the range of possible values for x and y. By taking into account the properties of the square root function and the definition of cos(C), we can deduce that the smallest possible value for cos(C) occurs when x and y are at their minimum values (0), and c is at its maximum value (√(x^2 + xy + y^2)). Thus, the largest angle will correspond to the value of cos(C) when x = y = 0 and c = √(x^2 + xy + y^2).

Substituting these values into the equation, we get:
cos(C) = (√(0^2 + 0 * 0 + 0^2)) / 2 * 0 * 0
= (√(0)) / 0
= 0 / 0

Since dividing by zero is undefined, we cannot determine the largest angle of the triangle with the given side lengths.