I have a caternary arch kiln. the inside base, is 4 feet. The apex of the inside parabolic arch is four feet. How many feet is the inside of the kiln?

To determine the length of the inside of the kiln, we need to calculate the distance between the two endpoints of the catenary arch. A catenary arch is a specific type of curve that is formed when a chain or cable is suspended between two points and allowed to hang freely under its own weight.

In this case, the length of the inside base of the kiln is given as 4 feet, which means that the distance between the endpoints of the catenary arch is also 4 feet. We can assume that the inside shape of the kiln follows a perfect catenary curve.

To find the distance between the endpoints, we can use the formula for the catenary curve:

y = a * cosh(x/a)

Where:
y represents the height of the arch at any given point
x represents the horizontal distance from the midpoint of the arch
a represents the constant that determines the shape of the curve

Since the apex of the inside parabolic arch is also at a height of 4 feet, we can substitute these values into the equation:

4 = a * cosh(0/a)

Simplifying the equation, we find:

cosh(0/a) = 4/a

Now, we need to solve this equation to find the value of "a" (the constant that determines the shape of the curve). This can be done through numerical methods or by using a scientific calculator with a hyperbolic cosine function (cosh).

Once you have determined the value of "a," you can substitute it back into the equation y = a * cosh(x/a) and calculate the height of the arch at any given distance x. The length of the inside of the kiln would then be obtained by measuring the total horizontal distance between the endpoints of the catenary arch.

Keep in mind that this calculation assumes a perfect catenary curve, and the actual shape of your kiln may differ slightly from this idealized model.