   Chapter 8.1, Problem 43E

Chapter
Section
Textbook Problem

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(πx/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.) To determine

To verify: The sine curve equation as y=sin(πx7).

To find: The width of a flat metal sheet to make a 28-inch panel.

Explanation

Given information:

The sine curve function is y=sin(πx7) (1)

The lower limit is 0 and the upper limit is 28inch.

Refer sine curve.

Sine wave covers two wave periods in the 28 inch distance.

The sine wave has amplitude of 1 and wave period of 14.

The sine wave is expressed as,

y=asin2πxT (2)

Here, the amplitude is a and the wave period is T.

Substitute 1 for a and 14 for T in Equation (2).

y=1sin2πx14=sin(πx7)

Thus the sine curve has the equation of y=sin(πx7).

Hence, the sine curve equation is verified.

The width (w) of the metal sheet is used to make the panel is considered as arc length of the sine curve.

The expression to find the arc length of the curve (L) is shown below:

L=ab1+(dydx)2dx (3)

Here, the derivative of the function y is dydx, the lower limit is a, and the upper limit is b

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