Hyperbolic and Inverse Hyperbolic Functions: Problem Solving

An arched gate can be effectively modeled using a hyperbolic cosine profile because this type of function is smooth and symmetric about the vertical axis. When the arch is centered at the origin, its maximum height occurs at the center point. This symmetry ensures that any height below the crown of the arch is reached at two horizontal positions that are equal in distance from the centerline but lie on opposite sides.

To determine where the gate reaches a height of five meters, the height of the function is adjusted, and the corresponding horizontal positions are identified. Solving this relationship shows that there are exactly two locations where the structure attains this height, reflecting the inherent symmetry of the hyperbolic cosine shape.

To construct the tangent lines at these locations, the steepness of the arch must be evaluated. This is done by finding how rapidly the height of the curve changes with respect to the horizontal direction. Evaluating this rate of change at the two positions where the height is five meters yields slopes that are equal in magnitude but opposite in sign. One slope is positive, indicating that the left side of the arch rises toward the center, while the other is negative, indicating that the right side descends away from the center.

Using each point on the arch together with its corresponding slope, tangent lines can be written that locally approximate the shape of the structure near the five-meter level. These tangent lines provide linear representations of the curve in the immediate vicinity of those points and are useful for estimating the inclination required for structural and engineering design.