The profile of the cables on a suspension bridge may be modeled by a parabola. The central span of the bridge is 1260 m long and 139 m high. The parabola y equals 0.00035 x squared gives a good fit to the shape of the​ cables, where StartAbsoluteValue x EndAbsoluteValue less than or equals 630​, and x and y are measured in meters. Approximate the length of the cables that stretch between the tops of the two towers.
1260 m
139 m
y
x
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Part 1
The length of the cables is approximately 
  
enter your response here meters.
​(Round to the nearest whole​ number.)

The length of the cables can be found by calculating the arc length of the parabola between the two towers, which is modeled by the equation \( y = 0.00035x^2 \), where \( |x| \leq 630 \). The formula for the arc length of a function \( y = f(x) \) over an interval \( [a, b] \) is:

\[
L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx
\]

Here, \( \frac{dy}{dx} \) is the derivative of \( y = 0.00035x^2 \), so we first compute the derivative:

\[
\frac{dy}{dx} = 2(0.00035)x = 0.0007x
\]

Now, we need to plug this into the arc length formula. We are computing the arc length from \( x = -630 \) to \( x = 630 \), so the total arc length \( L \) is:

\[
L = \int_{-630}^{630} \sqrt{1 + (0.0007x)^2} \, dx
\]

Let's calculate this integral to find the approximate length of the cables.The approximate length of the cables stretching between the tops of the two towers is 1300 meters (rounded to the nearest whole number).

Would you like further details or explanations on how this was calculated?

Here are five related questions you might find interesting:
1. How is the arc length formula derived from calculus?
2. What other shapes can suspension bridge cables follow, aside from parabolas?
3. Can the arc length formula be applied to 3D curves?
4. What are practical uses of arc length calculations in engineering?
5. How would the cable length change if the span or height of the bridge changed?

**Tip:** When calculating the arc length of a curve, always check if symmetry can simplify the computation.

The profile of the cables on a suspension bridge may be modeled by a parabola. The central span of the bridge is 1260 m long and 139 m high. The parabola y equals 0.00035 x squared gives a good fit to the shape of the cables, where |x| ≤ 630, and x and y are measured in meters. Approximate the length of the cables that stretch between the tops of the two towers.

This problem demonstrates how to calculate the length of suspension bridge cables modeled by a parabola using the arc length formula. The central span is 1260 meters long, 139 meters high, and the equation y = 0.00035x^2 is used to approximate the shape. Suitable for advanced high school students.

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