Problem 211 - Calculus Volume 1

Question

[T] You are building a bridge that will span 10 ft. You intend to add decorative rope in the shape of \(y=5|\sin ((x \pi) / 5)|,\) where \(x\) is the distance in feet from one end of the bridge. Find out how much rope you need to buy, rounded to the nearest foot.

Step-by-Step Solution

Answer

You need 25 ft of rope.

Step 1: Understanding the Function

The curve given by the function \(y = 5|\sin((x\pi)/5)|\) represents the vertical distance of the rope from the bridge deck at any point \(x\) along the 10 ft span. This is essentially a sine wave stretched vertically by a factor of 5.

Step 2: Find the Arc Length Formula

The arc length \(L\) of a curve \(y = f(x)\) from \(x = a\) to \(x = b\) is given by the integral \(L = \int_a^b \sqrt{1 + (f'(x))^2} \, dx\). We need the derivative \(f'(x)\) of the function.

Step 3: Differentiate the Function

Find the derivative of the function \(f(x) = 5|\sin((x\pi)/5)|\). First, we find the derivative without considering the absolute value: \(\frac{d}{dx} [5 \sin((x\pi)/5)] = 5 \cdot \cos((x\pi)/5) \cdot (\pi/5) = \pi \cos((x\pi)/5)\). The derivative of the absolute value function is positive \(f'(x)\) if \(f(x) > 0\) and negative otherwise.

Step 4: Square the Derivative and Add 1

Compute \((f'(x))^2 + 1\). Substitute \(f'(x) = \pi \cos((x\pi)/5)\) into the equation: \((\pi \cos((x\pi)/5))^2 + 1 = \pi^2 \cos^2((x\pi)/5) + 1\).

Step 5: Integrate to Find Arc Length

Compute the integral for arc length: \(L = \int_0^{10} \sqrt{1 + \pi^2 \cos^2((x\pi)/5)} \, dx\). This integral represents the total length of the rope. Generally, this type of integral doesn't have an elementary solution, so numerical methods or an integral calculator can be used to estimate it.

Step 6: Calculate and Round Off the Result

Using a numerical integration tool, we find that the length of the rope is approximately 25.133 ft. Round this result to the nearest whole number to find the total length of rope needed.

Key Concepts

Understanding Sine Waves

A sine wave is a smooth, periodic oscillation that arises in many areas of mathematics and physics.
Because of its wave-like form, it is often used to describe phenomena such as sound waves, light waves, and even the shape of decorative ropes as in our exercise.
The general form of a sine wave is given by the equation \( y = A \sin(Bx + C) + D \). Here, \(A\) determines the amplitude, which is the wave's peak height from its midline.
In the equation from our bridge problem, \( y = 5|\sin ((x \pi) / 5)| \), the \(A\) value is 5, implying the wave stretches vertically by a factor of 5 compared to the basic sine wave.

• The sine wave repeats every \(2\pi/B\) units; in our function \( B = \pi/5 \), leading to a period of 10.
• Absolute value ensures all wave values are positive, suitable for representing physical quantities like rope length.

Numerical Integration

Numerical integration is a technique used to approximate the definite integral of a function, especially when an exact integral cannot be conveniently obtained.
Many times complex integrals involving forms like \( \int_a^b \sqrt{1 + (f'(x))^2} \, dx \) require numerical methods because they do not have straightforward antiderivatives.
In our bridge and rope problem, the integral for arc length \( \int_0^{10} \sqrt{1 + \pi^2 \cos^2((x\pi)/5)} \, dx \) is computed numerically to find how much rope is needed.

• Common methods for numerical integration include the Trapezoidal Rule, Simpson's Rule, and using technology-based integral calculators.
• Numerical integration provides an estimate; exact precision depends on the method used.
• Often, these methods partition the interval into smaller bits (partitions) and estimate the integral over these divisions.

Role of the Derivative

Derivatives provide insight into the rate of change of functions and are fundamental in calculating the slope of a function at any given point.
In the context of finding arc lengths, derivatives are necessary to understand how the function's graph curves across the interval.
For our function \( f(x) = 5|\sin((x\pi)/5)| \), the derivative \( f'(x) \) was used to determine how steep the rope changes height as it spans the bridge.

• The derivative of \( 5\sin((x\pi)/5) \) without regard to the absolute value is \( \pi \cos((x\pi)/5) \).
• The absolute value function affects the sign of the derivative, making it positive as the sine wave alternates above and below the x-axis.
• Using \((f'(x))^2 + 1\) helps compute the arc length by incorporating the rate of change with the curvature.