Answered: 4. The profile of the cables of a…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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$**Title: Calculating the Length of Suspension Bridge Cables Using Definite Integrals** --- **Introduction:** The profile of the cables of a suspension bridge can be modeled by a parabola. In this lesson, we'll explore how to set up the definite integral to find the length of the cables for the central span of the Golden Gate Bridge. **Problem Statement:** Consider the central span of the Golden Gate Bridge, which is 1280 meters long and 152 meters high. The shape of the cables can be approximated by the parabolic equation $ y = 0.00037x^2 $, where $|x| \leq 640$ and both $x$ and $y$ are measured in meters. **Objective:** Find the definite integral that gives the length of the cables stretching between the tops of the two towers. We will set up the integral without solving it. **Parabolic Equation:** \[ y = 0.00037x^2 \] **Integration Setup:** To determine the length of the parabolic cable, we use the arc length formula for a function $ y = f(x) $: \[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \] **Step-by-Step Integral Setup:** 1. **Differentiate the Parabolic Equation:** First, find the derivative of $y$ with respect to $x$: \[ \frac{dy}{dx} = \frac{d}{dx}(0.00037x^2) = 0.00074x \] 2. **Compute $\left(\frac{dy}{dx}\right)^2$:** \[ \left(\frac{dy}{dx}\right)^2 = (0.00074x)^2 = 0.0005476x^2 \] 3. **Set Up the Integrand:** Substitute $\left(\frac{dy}{dx}\right)^2$ into the arc length formula: \[ \sqrt{1 + \left(\frac{dy}{dx}\right)^2} = \sqrt{1 + 0.0005476x^2} \] 4. **Define the Limits of Integration:** Since the span of the bridge is from $-640$ meters to $640$ meters:$

Transcribed Image Text:**Title: Calculating the Length of Suspension Bridge Cables Using Definite Integrals** --- **Introduction:** The profile of the cables of a suspension bridge can be modeled by a parabola. In this lesson, we'll explore how to set up the definite integral to find the length of the cables for the central span of the Golden Gate Bridge. **Problem Statement:** Consider the central span of the Golden Gate Bridge, which is 1280 meters long and 152 meters high. The shape of the cables can be approximated by the parabolic equation $ y = 0.00037x^2 $, where $|x| \leq 640$ and both $x$ and $y$ are measured in meters. **Objective:** Find the definite integral that gives the length of the cables stretching between the tops of the two towers. We will set up the integral without solving it. **Parabolic Equation:** \[ y = 0.00037x^2 \] **Integration Setup:** To determine the length of the parabolic cable, we use the arc length formula for a function $ y = f(x) $: \[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \] **Step-by-Step Integral Setup:** 1. **Differentiate the Parabolic Equation:** First, find the derivative of $y$ with respect to $x$: \[ \frac{dy}{dx} = \frac{d}{dx}(0.00037x^2) = 0.00074x \] 2. **Compute $\left(\frac{dy}{dx}\right)^2$:** \[ \left(\frac{dy}{dx}\right)^2 = (0.00074x)^2 = 0.0005476x^2 \] 3. **Set Up the Integrand:** Substitute $\left(\frac{dy}{dx}\right)^2$ into the arc length formula: \[ \sqrt{1 + \left(\frac{dy}{dx}\right)^2} = \sqrt{1 + 0.0005476x^2} \] 4. **Define the Limits of Integration:** Since the span of the bridge is from $-640$ meters to $640$ meters:

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