A suspension bridge with weight uniformly distributed along its length has twin towers that extend 50 meters above the road surface and are 1200 meters apart. The cables are parabolic in shape and are suspended from the tops of the towers The cables touch the road surface at the center of the bridge. Find the height of the cables at a point 300 meters from the center. (Assume that the road is level.) The height of the cables is meters. (Simplify your answer.)
A suspension bridge with weight uniformly distributed along its length has twin towers that extend 50 meters above the road surface and are 1200 meters apart. The cables are parabolic in shape and are suspended from the tops of the towers The cables touch the road surface at the center of the bridge. Find the height of the cables at a point 300 meters from the center. (Assume that the road is level.) The height of the cables is meters. (Simplify your answer.)
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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![### Problem Statement: Suspension Bridge Cable Height Calculation
**Scenario:**
A suspension bridge with weight uniformly distributed along its length has twin towers that extend 50 meters above the road surface and are 1200 meters apart. The cables are parabolic in shape and are suspended from the tops of the towers. The cables touch the road surface at the center of the bridge.
**Objective:**
Find the height of the cables at a point 300 meters from the center of the bridge. (Assume that the road is level.)
**Mathematical Model:**
1. The distance between the two towers: 1200 meters.
2. Height of the towers above the road surface: 50 meters.
3. The center of the bridge is the lowest point of the cable, which touches the road surface.
**Question:**
The height of the cables is meters.
**Instruction:**
(Simplify your answer.)
---
In this scenario, assume a bridge represented by a parabolic equation. Let's denote:
- The horizontal distance from the center of the bridge to any point on the cable as \( x \),
- The vertical distance from the road to the cable at any point as \( y \).
The vertex of the parabola is at the center of the bridge, where the cable touches the road surface (vertex at point (0,0)). To determine the height of the cables at a point 300 meters from the center, we need to find the value of \( y \) when \( x = 300 \).
Given:
- The towers are 600 meters away from the center (half the total span).
- The height of the cable at the towers is 50 meters.
The general equation for the parabola is:
\[ y = ax^2 \]
We know \( a \) by using the condition at the towers:
\[ 50 = a(600)^2 \]
\[ 50 = 360000a \]
\[ a = \frac{1}{7200} \]
Now, we find the height at 300 meters from the center:
\[ y = \frac{1}{7200}(300)^2 \]
\[ y = \frac{1}{7200}(90000) \]
\[ y = 12.5 \]
Therefore:
The height of the cables at a point 300 meters from the center is **12.5 meters**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F49360a23-9f24-445c-a223-e7141e33e2c6%2F40d7e6b6-b0cb-452c-a677-33d0fa271ef4%2Fojc4vef_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement: Suspension Bridge Cable Height Calculation
**Scenario:**
A suspension bridge with weight uniformly distributed along its length has twin towers that extend 50 meters above the road surface and are 1200 meters apart. The cables are parabolic in shape and are suspended from the tops of the towers. The cables touch the road surface at the center of the bridge.
**Objective:**
Find the height of the cables at a point 300 meters from the center of the bridge. (Assume that the road is level.)
**Mathematical Model:**
1. The distance between the two towers: 1200 meters.
2. Height of the towers above the road surface: 50 meters.
3. The center of the bridge is the lowest point of the cable, which touches the road surface.
**Question:**
The height of the cables is meters.
**Instruction:**
(Simplify your answer.)
---
In this scenario, assume a bridge represented by a parabolic equation. Let's denote:
- The horizontal distance from the center of the bridge to any point on the cable as \( x \),
- The vertical distance from the road to the cable at any point as \( y \).
The vertex of the parabola is at the center of the bridge, where the cable touches the road surface (vertex at point (0,0)). To determine the height of the cables at a point 300 meters from the center, we need to find the value of \( y \) when \( x = 300 \).
Given:
- The towers are 600 meters away from the center (half the total span).
- The height of the cable at the towers is 50 meters.
The general equation for the parabola is:
\[ y = ax^2 \]
We know \( a \) by using the condition at the towers:
\[ 50 = a(600)^2 \]
\[ 50 = 360000a \]
\[ a = \frac{1}{7200} \]
Now, we find the height at 300 meters from the center:
\[ y = \frac{1}{7200}(300)^2 \]
\[ y = \frac{1}{7200}(90000) \]
\[ y = 12.5 \]
Therefore:
The height of the cables at a point 300 meters from the center is **12.5 meters**.
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