Final answer:
The intersection point of the missile with the y-axis can be found by determining the slope of the tangent to the given ellipse at (3, 8/5), forming the equation of the tangent line, and then solving for the y-intercept when x is set to 0.
Step-by-step explanation:
To find where the missile intersects the y-axis, we need to determine the equation of the tangent to the given ellipse at the point (3, 8/5). The ellipse provided is x² + 6.25y² = 25. To find the tangent, we can use the derivative as a slope at a given point and then write the equation of the line. The derivative of the ellipse equation is not given, but we can implicitly differentiate the equation of the ellipse. As we're only interested in the y-intercept of the tangent, it's sufficient to find the slope of the ellipse at the point in question.
The slope of the tangent line to the ellipse at any point (x, y) can be calculated using the implicit differentiation of the ellipse's equation. Differentiating both sides with respect to x gives 2x + (6.25 × 2y × y') = 0, which simplifies to y' = -2x / (6.25 × 2y). At the point (3, 8/5), the slope (y') is -3 / (6.25 × 8/5) = -15 / (6.25 × 8). The equation of the tangent line can be written as y - 8/5 = m(x - 3), where m is the slope of the tangent at (3, 8/5). Plugging the values in, we get the equation y - 8/5 = (-15 / (6.25 × 8))(x - 3).
Next, to find the y-intercept, set x to 0 and solve for y: y = -15 / (6.25 × 8) × (-3) + 8/5. After simplifying, this will give us the point at which the missile will intersect the y-axis. This value is the y-intercept of the tangent, and therefore, the intersection point of the missile with the incoming asteroid.