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A cable hangs between two poles of equal height and 35 feet apart. At a point on the ground...

Question:

A cable hangs between two poles of equal height and {eq}35 {/eq} feet apart. At a point on the ground directly under the cable and {eq}x {/eq} feet from the point on the ground halfway between the poles the height of the cable in feet is {eq}h(x)=10+(0.5)x^{1.5} {/eq}. The cable weighs {eq}18.3 {/eq} pounds per linear foot. Find the weight of the cable.

Arc Length:

To find the length of the curve {eq}f(x) {/eq} over the interval {eq}[a,b] {/eq}, we can use the following arc length integral formula:

{eq}\displaystyle \int_{a}^{b} \sqrt{1 +(f'(x))^2)} dx {/eq}

Here {eq}f'(x) {/eq} is the first derivative of {eq}f(x) {/eq}. Once we have our definite integral set up, we can evaluate it using standard integration techniques, such as u-substitution.

Answer and Explanation:

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Given Data:

  • The distance between the two poles is: {eq}d = 35\;{\rm{ft}} {/eq}
  • The height of the cable is: {eq}h\left( x \right) = 10 + 0.5\left(...

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How to Find the Arc Length of a Function

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Chapter 12 / Lesson 12
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