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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