2.22: 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.

Consider an arched gate that follows a precise hyperbolic cosine function.

Since the curve is symmetrical, evaluating the function at the origin provides the maximum height.

The task is to identify the positions where the height reaches 5 meters and compute the tangent line equations at these positions to estimate the slope required for structural design.

Finding where the structure horizontally reaches a given height requires adjusting the function value.

Solving for the horizontal positions identifies two locations that satisfy this condition.

Now, to compute the tangent line equations, the first step is to find the steepness of the curve at these specific positions.

To do this, differentiate the function step by step and calculate the derivative of the function at these points.

Substituting the horizontal position into the derivative equation gives slopes of about +0.98 and –0.98, reflecting the structure's symmetry.

Next, consider the point-slope form of a line. Substitute these slopes and the corresponding coordinates to obtain the equations of the tangent lines.

This shows the application of hyperbolic functions in calculus and real-world modeling.