Hyperbolic and Inverse Hyperbolic Functions: Problem Solving

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.