Question

A cable hangs between two poles of equal height and 35 feet apart. At a point on the ground directly under the cable and x feet from the point on the ground halfway between the poles the height of the cable in feet is h(x)=10+(0.1)(x1.5). The cable weighs 10.4 pounds per linear foot. Find the weight of the cable.

Answer 1

293.38 pounds

Explanation:

We are given that

Distance between poles=35 feet

`h(x)=10+0.1(x^(1.5))`

Weight of cable=10.4 per linear foot

We have to find the weight of the cable.

Differentiate w.r.t

`h'(x)=0.1(1.5)x^(0.5)=0.15x^(0.5)`

`s=2\int_(0)^(17.5)√(1+(h'(x))^2)dx`

`s=2\int_(0)^(17.5)\sqrt{1+(0.15x^(0.5))^2}dx`

`s=2\int_(0)^(17.5)√(1+0.0225x)dx`

Let
`1+0.0225x=t`

`dx=(1)/(0.0225)dt`

`s=(2)/(0.0225)\int_(0)^(17.5)√(t)dt`

`s=(2)/(0.0225)*(2)/(3)[t^{(3)/(2)}]^(17.5)_(0)`

`s=2* (2)/(3*0.0225)[(1+0.0255x)^{(3)/(2)]^(17.5)_(0)`

`s=(4)/(3* 0.0225)((1+0.0225(17.5))^{(3)/(2)-1)`

s=28.21

Weight of cable=
`28.21* 10.4=293.38`pound