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+(0.1)*(x^1.5). The cable weighs 19.2 pounds per linear foot. Find the weight of the cable.

Question

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+(0.1)*(x^1.5). The cable weighs 19.2 pounds per linear foot. Find the weight of the cable.

Answer 1

h' = 3/20 √x

the length of cable is

s = 2∫[0,20] √(1+h'^2) dx
= 2∫[0,20] √(1+9x/400) dx
= 44.2 ft

Now figure the weight.

Answer 2

To find the weight of the cable, we need to determine the length of the cable first.

Since the cable hangs between two poles of equal height and 40 feet apart, we can imagine a right triangle formed by the cable, the ground, and a line connecting the midpoint of the cable to the top of one of the poles. The height of the triangle (length of the cable) can be determined using the Pythagorean theorem.

Let's call the length of the cable L. The height of the triangle (h) can be calculated using the equation:

h^2 = (40/2)^2 + L^2

Simplifying this equation, we get:

h^2 = 20^2 + L^2

Now, let's solve for L:

L^2 = h^2 - 20^2

Since h(x) represents the height of the cable above the ground at a certain point x, we need to find the average height of the cable from the left pole to the right pole. This can be calculated by integrating h(x) from 0 to 40 (the entire distance between the poles) and dividing by the distance:

Average height = (1/(40 - 0)) * ∫[0 to 40] h(x) dx

Let's perform the integration:

Average height = (1/40) * ∫[0 to 40] (10 + 0.1 * (x^1.5)) dx

Average height = (1/40) * [(10x + (0.1 * (2/3) * x^(1.5+1))) | from 0 to 40]

Average height = (1/40) * [(10 * 40 + (0.1 * (2/3) * 40^(1.5+1))) - (10 * 0 + (0.1 * (2/3) * 0^(1.5+1)))]

Average height = (1/40) * [400 + (0.1 * (2/3) * 40^2.5)]

Next, we will substitute the average height back into the equation to solve for L:

h^2 = 20^2 + L^2

[(1/40) * [400 + (0.1 * (2/3) * 40^2.5)]]^2 = 20^2 + L^2

Now we can find L by solving this equation:

L^2 = [(1/40) * [400 + (0.1 * (2/3) * 40^2.5)]]^2 - 20^2

Finally, to find the weight of the cable, we multiply the length of the cable (L) by the weight per linear foot (19.2 pounds per linear foot):

Weight of the cable = L * 19.2 pounds per linear foot