Problem 67 - Calculus of a Single Variable

Question

Suspension Bridge A cable for a suspension bridge has the shape of a parabola with equation \(y=k x^{2} .\) Let \(h\) represent the height of the cable from its lowest point to its highest point and let 2\(w\) represent the total span of the bridge (see figure). Show that the length \(C\) of the cable is given by $$ C=2 \int_{0}^{w} \sqrt{1+\left(4 h^{2} / w^{4}\right) x^{2}} d x $$

Step-by-Step Solution

Answer

The length of the parabolic cable for the suspension bridge is given by \(C=2\int_{0}^{w} \sqrt{1+\left(4 h^{2} / w^{4}\right) x^{2}} d x\). This is derived by first finding the derivative of the function, squaring it and then substituting it in the arc length formula.

Step 1: Find derivative of the function

First, let's find the derivative of the function \(y=k x^{2}\). The derivative of this function with respect to \(x\) is \(y' = 2kx\). Now, we know that \(k=h/w^{2}\) since at \(x=w\), \(y=h\). So, \(y' = 2hx/w^{2}\). Square of this equation is \((y')^{2}=(4h^{2}x^{2})/(w^{4})\).

Step 2: Apply arc length formula

The formula to calculate arc length is \(C=\int_{a}^b \sqrt{1+(y')^{2}} dx\). Substituting the expression of \((y')^{2}\) and \(a=0\), \(b=w\) in the formula, we get, the length \(C\) of the cable is \(C=2\int_{0}^{w} \sqrt{1+\left(4 h^{2} / w^{4}\right) x^{2}} d x\). This formula gives us the length of a parabolic cable of a suspension bridge.

Key Concepts

Arc Length

In calculus, the arc length of a curve is the distance measured along the path of the curve. It is a way to determine the total length of a curve between two points. To find the arc length of a curve defined by a function, we use an integral formula.

• The general formula for the arc length of a curve from point \(a\) to \(b\) is given by: \[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]
• This involves integrating the square root of the sum of 1 and the square of the derivative of the function.
• It allows us to account for the curve's bending, not just its horizontal change.

In our exercise, this concept is used to determine the length of the cable of a suspension bridge shaped like a parabola. By integrating over the span of the bridge, we obtain the precise cable length.

Parabola

A parabola is a symmetrical, curved shape that is often encountered in mathematics, especially in algebra and calculus. It is the graph of a quadratic function, which can be expressed in the form \(y = ax^2 + bx + c\).

• In the case of a suspension bridge, the cable naturally forms a parabolic shape due to the forces acting on it, such as gravity.
• In the exercise, the parabola is defined by the equation \(y = kx^2\), where \(k\) is a scaling factor that influences the curvature.
• The vertex of the parabola—its lowest point—is at the origin when \(x=0\).

Understanding the nature of a parabolic curve is essential for calculating the arc length, as it allows us to properly set up the integral formula needed to find the cable length.

Derivative

Derivatives in calculus represent the rate of change of a function regarding its variable. In simpler terms, the derivative shows how a function changes at any given point.

• For the equation \(y = kx^2\), the derivative \(y'\) is \(2kx\). This tells us how steep the parabola is at any point.
• The importance of the derivative in this problem lies in its use in the arc length formula. It helps describe the shape of the parabola precisely.
• Correctly finding the derivative of the function is crucial, as it forms part of the arc length's integrand.

By plugging the derivative into the arc length formula, we account for the changes in \(y\) as \(x\) travels from \(0\) to \(w\), ensuring an accurate measure of the curve's total length.

Suspension Bridge

Suspension bridges are marvels of engineering that rely on cables hanging in a parabolic curve to support the bridge deck below.

• The cables are under tension, pulling upwards and downwards at the same time, allowing the structure to support heavy loads.
• The exercise focuses on understanding the parabolic shape because it is the most efficient curve for distributing the load evenly along the cables.
• Using calculus, particularly the arc length formula, engineers can precisely calculate the cable's length, crucial for construction.

The blend of mathematics and engineering principles helps design suspension bridges that are not only strong but also aesthetically pleasing. Calculus, in particular, plays a fundamental role in ensuring that these structures perform safely and effectively.