Final answer:
The eccentricity of the ellipse 15x^2 + 9y^2 = 30 is sqrt(2/3).
Step-by-step explanation:
The eccentricity of the ellipse can be calculated using the formula (e) = fla, where f is the distance from the center of the ellipse to one of the foci and a is half the length of the major axis. In this case, the given equation of the ellipse is 15x^2 + 9y^2 = 30. To determine the eccentricity, we need to find the values of f and a.
First, we need to rewrite the equation in the standard form of an ellipse, which is (x^2/a^2) + (y^2/b^2) = 1. Dividing both sides by 30, we get (x^2/2) + (y^2/10/3) = 1. Comparing this equation with the standard form, we can see that a^2 = 2 and b^2 = 10/3.
Since a = sqrt(2), the semimajor axis is sqrt(2). To find the value of f, we can use the equation e^2 = 1 - (b^2/a^2). Substituting the values, we get e^2 = 1 - (10/3/2) = 1 - (5/3) = 2/3. Taking the square root of both sides, we get e = sqrt(2/3).