Question
- The profile of the cables of a suspension bridge may be modeled by a parabola. The centr
span of the Golden Gate Bridge is 1280 meters long and 152 meters high. The parabola
gives a good fit to the shape of the cables, where
, and
and
are
measured in meters. Find the definite integral that gives the length of the cables that
stretch between the tops of the two towers (set-up but do not solve).
span of the Golden Gate Bridge is 1280 meters long and 152 meters high. The parabola
measured in meters. Find the definite integral that gives the length of the cables that
stretch between the tops of the two towers (set-up but do not solve).
Ask by Carlson Whittaker. in the United States
Jan 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The length of the cables can be found using the integral:
Solution
To find the length of the cables modeled by the parabola
between the two towers of the Golden Gate Bridge, we’ll use the formula for the arc length of a curve defined by a function
.
Arc Length Formula:
Step-by-Step Setup:
-
Identify the Function and Its Derivative:
-
Square the Derivative:
-
Set Up the Integral Limits:
The span is frommeters to meters. -
Write the Definite Integral for Arc Length:
Final Integral Expression:
This integral represents the total length of the suspension bridge cables between the two tower tops. To find the numerical value of the length, you would evaluate this definite integral.
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Beyond the Answer
To find the length of the cables stretching between the tops of the two towers of the Golden Gate Bridge, we need to set up the definite integral for the arc length of the parabola described by the function
. The formula for the arc length
of a curve from
to
is given by:
First, we need to find
for our parabola:
Now substitute this into the arc length formula. If we are considering the length of the cables from the left tower at
meters to the right tower at
meters, our definite integral will be:
This integral represents the total length of the cables stretching between the tops of the two towers.