# Tilted Ellipsoid

Calculus Level 5

An ellipsoid is centered at the origin $$(0,0,0)$$. Its semi-axes are initially aligned with the $$xyz$$-axes. The semi-axis along the x-axis is 10 units, and the one aligned with the y-axis is 15 units, and the semi-axis aligned with the z-axis is 25.

Next, a coordinate frame $$x' y' z'$$ is rigidly attached to the ellipsoid, and initially aligned with the universal $$xyz$$ orientation.

Next, the ellipsoid is rotated clockwise about its $$x'$$-axis by $$60^\circ$$, then about its $$y'$$-axis clockwise by 45 degrees, and finally about its $$z'$$-axis clockwise by 30 degrees.

Find the $$z$$-coordinate (i.e. height) of the peak point of the rotated ellipsoid, correct to three decimal places.

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