Question

An elliptical arch is constructed which is 8 feet wide at the base and 8 feet tall in the middle. Find the height of the arch exactly 1 foot in from the base of the arch.

Answer 1

According to the data you gave, the equation of the ellipse will be
x^2/16 + y^2/64 = 1
or
64x^2 + 16y^2 = 1024
which reduces to
4x^2 + y^2 = 64
consider only the part above the x-axis

not clear what you mean by "1 foot from the base of the arc"
I will assume you mean when x = 3
36 + y^2 = 64
y^2 = 28
y = height of the arch = √28 = 2√7 or about 5.29 ft