A hallway 5.0 m wide has a ceiling whose cross section is a semi ellipse. The ceiling is 3.0 m high at the walls and 4.0 m high at the center. Find the height of the ceiling 1.0 m from each wall.
How would I start this problem?
From the measures of the hallway you know about the ellipse:
semi-major axis : 2.5 m
semi-minor axis : 1 m
coordinates of the center: C(0, 3)
Thus the equation of the ellipse is:
$\displaystyle \frac{x^2}{\left(\frac52\right)^2}+\frac{(y-3)^2}{1^2} = 1$
Because you only need the upper part of the ellipse you can solve for y to get the equation of a function:
$\displaystyle e(x) = 3+\sqrt{1-\frac{4x^2}{25}}$
The points A and B are 1 m from the wall: A(-1.5, e(1.5)) and B(1.5, e(1.5))
Because the ellipse is symmetric about the y-axis it is only necessary to calculate e(1.5).
I've got A(-1.5, 3.8), B(1.5, 3.8)