A cable hangs between two poles of equal height and 40 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+(0.4)(x^1.5).
The cable weighs 18.2 pounds per linear foot.
Find the weight of the cable.
You want the arc length of the cable. Just figure the length from the center and double it.
dh/dx = 0.6√x
so, the arc length from the center to the pole is just
∫[0,20] √(1+.36x) dx
= 50/27 (1+.36x)^(3/2) [0,20]
= 41.63
So, the cable weighs 18.2*2*41.63 = 1515.3 lbs
thank u
To find the weight of the cable, we need to calculate its total length and multiply it by the weight per linear foot.
First, let's find the equation of the cable's shape between the two poles.
Given that the height of the cable is given by the equation h(x) = 10 + 0.4(x^1.5), where x is the distance from the point on the ground directly under the cable and x is measured in feet.
The length of the cable between the two poles is equal to the integral of the square root of the sum of the squares of the derivatives of x and h(x).
Let's calculate it step by step:
1. The derivative of h(x) with respect to x is:
h'(x) = 0.6x^0.5
2. The square of the derivative is:
(h'(x))^2 = (0.6x^0.5)^2 = 0.36x
3. The integral of the square root of the sum of the squares of x and h'(x) is:
L(x) = ∫√(1 + (h'(x))^2) dx = ∫√(1 + 0.36x) dx
4. To find the limits of integration, we know that the cable is 40 feet long, so the limits of integration are from -20 to 20 (half of the distance between the poles).
L(x) = ∫[from -20 to 20] √(1 + 0.36x) dx
5. Evaluating the integral, we get:
L(x) = 2√(1 + 0.36x) |[from -20 to 20]
= 2(√(1 + 0.36(20)) - √(1 + 0.36(-20)))
6. Simplifying the expression:
L(x) = 2(√8.4 - √0.4)
= 2(2.897 - 0.632)
≈ 4.53 feet
Now we have the length of the cable, which is approximately 4.53 feet.
Finally, to find the weight of the cable, we multiply the length of the cable by the weight per linear foot:
Weight = Length × Weight per linear foot
= 4.53 feet × 18.2 pounds per linear foot
≈ 82.596 pounds
The weight of the cable is approximately 82.596 pounds.
To find the weight of the cable, we need to find the length of the cable first.
We are given that the distance between the two poles is 40 feet. Since the cable hangs between the two poles, its length is equal to the distance between the two poles.
Therefore, the length of the cable is 40 feet.
Now, we can find the weight of the cable by multiplying its length by the weight per linear foot.
The weight of the cable is:
Weight = Length x Weight per linear foot
Weight = 40 feet x 18.2 pounds per linear foot
Weight = 728 pounds
Therefore, the weight of the cable is 728 pounds.