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.
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
To find the height of the elliptical arch 1 foot in from the base, we can use the equation of an ellipse.
An ellipse is defined by its major radius (the distance from the center to the outermost point on the top or bottom) and its minor radius (the distance from the center to the outermost point on the left or right). In this case, the major radius is 8 feet (half of the width at the base), and the minor radius is 4 feet (half of the height at the middle).
The equation of an ellipse centered at the origin (0,0) is:
x^2/a^2 + y^2/b^2 = 1
Where a is the major radius and b is the minor radius.
Plugging in the values for this particular elliptical arch, we get:
x^2/8^2 + y^2/4^2 = 1
Simplifying the equation, we get:
x^2/64 + y^2/16 = 1
To find the height 1 foot in from the base, we need to find the value of y when x = 7 (since the base is 8 feet wide and we want to go 1 foot in from each side).
Plugging in x = 7 into our equation, we get:
7^2/64 + y^2/16 = 1
Simplifying further:
49/64 + y^2/16 = 1
Rearranging the equation to solve for y^2:
y^2/16 = 1 - 49/64
y^2/16 = 15/64
Now, we can cross multiply to isolate y^2:
64 * y^2 = 16 * 15
64y^2 = 240
Dividing both sides by 64:
y^2 = 240/64
y^2 = 15/4
Taking the square root of both sides, we get:
y = ±√(15/4)
y = ±(√15/2)
Since the height of the arch is positive, we can take the positive square root:
y = √(15/4)
Simplifying the square root:
y = √15/2
Therefore, the height of the elliptical arch exactly 1 foot in from the base is √15/2 feet.