Q.The ceiling in a hallway 20 ft wide is in the shape of a semi ellipse and 18 ft high at the centre. Find the height of the ceiling 4 feet from either wall if the height of the side walls is 12 ft.
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Start your 14-day free trial to unlock the full solution →Set up the semi-ellipse with its major axis along the 20 ft width and its extra rise (above the 12 ft walls) as the semi-minor axis, then evaluate the ellipse equation 4 ft in from a wall.
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Set up coordinates. Take the origin at the midpoint of the floor level directly below the centre of the ceiling, with measured horizontally across the 20 ft width and measured vertically above the top of the side walls (i.e. above height 12 ft). The hallway is 20 ft wide, so the semi-ellipse spans to ; thus the semi-major axis is .
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Find the semi-minor axis. The ceiling is 18 ft high at the centre and the side walls are 12 ft high, so the ellipse's extra rise above the wall level at the centre is ft. The cross-section of the ceiling (relative to the wall-top level) satisfies
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Locate the point 4 ft from a wall. A wall is at (or ); a point 4 ft from that wall is at .
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Solve for at . …
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