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.