A manufacturer of corrugated metal roofing wants to produce...

A manufacturer of corrugated metal roofing wants to produce panels that are 28 in. wide and 2 in. high by processing flat sheets of metal as shown in the figure. The profile of the roofing takes the shape of a sine wave. Verify that the sine curve has equation $ y = \sin (\frac{\pi x}{7}) $ and find the width $ w $ of a flat metal sheet that is needed to make a 28-inch panel. (Use your calculator to evaluate the integral correct to four significant digits.)

Key Concepts

Sine Function Transformation

This concept involves scaling and translating the basic sine function to meet specific design parameters. In the problem, the form y = sin(?x/7) is used so that the sine curve attains the desired amplitude and period. Specifically, by comparing the coefficient inside the sine, one shows that the amplitude is 1 (yielding a total vertical distance of 2 inches when measured from peak to trough) and the period is 14 inches, meaning half a period (7 inches) corresponds to going from 0 to the maximum, which fits the design of the roofing panel.

Arc Length of a Curve

The arc length concept is central to determining how long a flat sheet must be so that when it is corrugated into a sine-wave shape, the resulting panel has the correct horizontal width. This is computed using the formula L = ?[a to b] sqrt(1 + (dy/dx)²) dx, where dy/dx is the derivative of the sine function. The evaluation of this integral gives the length of the flat sheet needed before it is bent into the sine profile.

Numerical Integration

Since the integral resulting from the arc length formula often does not have a simple antiderivative, numerical integration techniques become necessary. In this problem, the integral is evaluated using a calculator to four significant digits. This emphasizes the practical need for numerical methods in applied problems where exact analytical answers are either unwieldy or unavailable.

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