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

A manufacturer of corrugated metal roofing wants to produc 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 signifi- cant digits.)

A manufacturer of corrugated metal roofing wants to produc
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 signifi-
cant digits.)

Key Concepts

Arc Length of a Curve

This concept involves finding the length of a curve described by a function over a given interval using the formula L = ?[a, b] ?(1 + (dy/dx)²) dx. In problems like the one given, the formula is used to determine how much raw material is needed when the curved shape is produced from a flat sheet, thereby requiring the calculation of the actual length of the sine-wave profile.

Properties of the Sine Function

The sine function is characterized by its amplitude, period, and phase. In the context of the roofing panel problem, the given sine curve y = sin(?x/7) is chosen because its amplitude determines the thickness of the corrugation (with a total height of 2 inches, corresponding to an amplitude of 1 inch), while the frequency (or period) of the sine function is adjusted to match the panel width. Verifying the equation involves checking that these features meet the design specifications.

Definite Integration

Definite integration is used to compute the exact length of a curve between two specified points. In this problem, after finding the derivative of the sine function, a definite integral is set up over one or more periods of the sine wave to calculate the required width of the flat metal sheet. The evaluation of this integral, typically using numerical methods for functions that do not have simple antiderivatives, ensures that precise measurements are obtained.

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