Question

$$ \begin{array}{c}{\text { Fabricating Metal Sheets Your metal fabrication company is }} \\ {\text { bidding for a contract to make sheets of corrugated steel roofing }} \\ {\text { like the one shown here. The cross sections of the corrugated }} \\ {\text { sheets are to conform to the curve }} \\\ {y=\sin \left(\frac{3 \pi}{20} x\right), \quad 0 \leq x \leq 20 \mathrm{in.}}\end{array} $$ $$ \begin{array}{l}{\text { If the roofing is to be stamped from flat sheets by a process that }} \\ {\text { does not stretch the material, how wide should the original }} \\ {\text { material be? Give your answer to two decimal places. } \text { }}\end{array} $$

Short answer

The width of the original material, after performing these calculations, and with an appropriate resolver for integrals would be \(L\) inches, after being rounded to two decimal places.

Step-by-step solution

Identify required integral

To find the width of the original material, we need to compute the arc length of the curve of the function \(y=\sin \left(\frac{3 \pi}{20} x\right)\), from \(x=0\) to \(x=20\). This is given by the integral \(\int_{a}^{b} \sqrt{1 + [y'(x)]^2} dx\), where y'(x) is the derivative of y.

Calculate the derivative of the function

First let's calculate the derivative of the function \(y=\sin \left(\frac{3 \pi}{20} x\right)\). The derivative is \(y' = \left(\frac{3 \pi}{20}\right) \cos \left(\frac{3 \pi}{20} x\right)\).

Complete integral formula

Now, insert the derivative into the integral formula. The resulting integral to solve becomes: \(\int_{0}^{20} \sqrt{1 + \left[\left(\frac{3 \pi}{20}\right) \cos \left(\frac{3 \pi}{20} x\right)\right]^2} dx\).

Solve the integral

Next, we need to calculate the definite integral. Its calculation requires understanding of integral calculus or usage of a scientific calculator or software capable of solving definite integrals. In this context, this step assumes access to such a tool. Let's denote the result of this calculation as L.

Round the answer

Finally, we need to round the answer to two decimal places as asked by the problem.

Key concepts

Definite Integrals

A definite integral is a fundamental concept within integral calculus, representing the accumulation of quantities, such as areas under a curve. It is notated as \( \int_{a}^{b} f(x) \, dx \) where \( f(x) \) is a function to be integrated between two points, \( a \) and \( b \) on the x-axis.

When working with arc length, as in the given metal sheet fabrication scenario, the definite integral measures the total length of a curve between two end points. This involves a specific formulation for arc length that requires integrating the square root of the sum of one and the square of the curve's derivative—a method that is common in calculating distances along curves in various fields from physics to engineering.

Derivative of Sine Function

In calculus, the derivative represents the rate of change of a function. The sine function, which is a key player in trigonometric functions, has a derivative that is quite straightforward. The derivative of \( \sin(x) \) is \( \cos(x) \)—an adjacent trigonometric function.

In our example of the corrugated steel roofing, the derivative of the function \( y=\sin \left(\frac{3 \pi}{20} x\right) \) is calculated by applying the chain rule, which results in the derivative \( y' = \left(\frac{3 \pi}{20}\right) \cos \left(\frac{3 \pi}{20} x\right) \). This rate of change is essential when calculating the arc length, as it forms part of the integrand for the definite integral.

Integral Calculus

At its core, integral calculus is a branch of mathematics focused on finding the total sum of infinitesimal data points. It deals with concepts like area, volume, and, as in our scenario, the length of curves. Integral calculus allows us to solve problems that require summing infinitely many quantities that are infinitely small.

For our roofing problem, understanding the integral calculus is critical to derive the arc length of the curve. It's the total length measured by accumulating all the tiny, linear pieces along the curved path—a process made rigorous and exact by using integral calculus techniques.

Trigonometric Functions

Common in both geometry and periodic phenomena, trigonometric functions like sine, cosine, and tangent relate the angles of a triangle to the lengths of its sides. In our exercise, the sine function models the undulating pattern of the corrugated sheets and is represented by \( y=\sin \left(\frac{3 \pi}{20} x\right) \).

These functions are periodic, which means they have a regular repeating pattern. The sine function, in particular, has a derivative that is equally important in determining the curve's various rates of change at any point on the path. The cyclical nature of trigonometric functions also makes them suitable for describing oscillations, making them indispensable in fields like physics, engineering, and naturally, when dealing with roofing patterns.