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. thick 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 $y=\sin (\pi x / 7)$ 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.)

A manufacturer of corrugated metal roofing wants to
produce panels that are 28 in. wide and 2 in. thick 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 $y=\sin (\pi x / 7)$ 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.)

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Key Concepts

Sine Function and its Properties

The sine function is a periodic function that models oscillatory behavior. Here, verifying the equation y = sin(?x/7) involves ensuring that the amplitude and period match the physical dimensions of the corrugated roofing. Specifically, the coefficient inside the sine function determines the period of the wave, which is crucial for designing the panel to have a specific width, while the amplitude correlates to the thickness of the panel.

Arc Length of a Curve

The arc length of a curve is the distance along the curve between two points, calculated using the formula L = ??(1 + (dy/dx)²) dx. In this context, the arc length represents the length of the flat metal sheet needed to form the sinusoidal corrugation pattern. Correctly setting up and evaluating this integral is essential to determine the material required.

Definite Integration and Numerical Approximation

Definite integration is used to evaluate the arc length over a fixed interval, in this case corresponding to the width of the panel. When an integral does not have an elementary antiderivative, numerical methods or a calculator are used to approximate the value. Here, the integral must be computed to four significant digits to ensure engineering accuracy in determining the necessary width of the flat metal sheet.

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