Final answer:
The area of the ellipse described by the equation 25x² + 36y² = 900, with substitutions x = 6u and y = 5v, is 30π square units after identifying the semi-major and semi-minor axes.
Step-by-step explanation:
To find the area of the region bounded by the ellipse with the equation 25x² + 36y² = 900, we must first identify the semi-major and semi-minor axes. Using the substitutions given, where x = 6u and y = 5v, we can rewrite the equation of the ellipse in terms of u and v:
25(6u)² + 36(5v)² = 900
Now, simplifying this,
900u² + 900v² = 900
Which can be further simplified to:
u² + v² = 1
From the last equation, it's clear that this represents an ellipse with a semi-major axis, a, of 1 (since u is the 'x' component) and a semi-minor axis, b, of 1 (since v is the 'y' component), but these need to be scaled back up by the substitutions given (x = 6u and y = 5v) to find the original dimensions of the ellipse.
Therefore, the semi-major axis of the original ellipse is a = 6 * 1 = 6 and the semi-minor axis is b = 5 * 1 = 5.
The area A of the ellipse is given by the formula A = πab, so subbing in our values for a and b:
A = π * 6 * 5
A = 30π
Thus, the area of the ellipse is 30π square units.