A vertical pole 24ft. high is to be braced securely by four wires attached 8ft. from the top. If the wires are fastened to stakes in the level ground at the vertices of square with its center at the foot of the pole and sides 20ft., what should be the length of each wire?.

Pythagorean Theorem:

16^2 + 20^2 = x^2

To find the length of each wire, we can use the Pythagorean Theorem. Let's consider one of the wires as the hypotenuse of a right triangle, with the pole as the height and the distance from the stake to the foot of the pole as the base.

We know that the pole is 24ft high and the wires are attached 8ft from the top of the pole. So, the height of the right triangle is 24ft - 8ft = 16ft.

Now, let's consider the square formed by the stakes on the ground. The sides of the square are 20ft, and the diagonal of the square is the base of the right triangle formed by the wire.

Using the Pythagorean Theorem, we can calculate the length of the diagonal of the square, which is the base of the right triangle:

diagonal^2 = side^2 + side^2
diagonal^2 = 20ft^2 + 20ft^2
diagonal^2 = 400ft^2 + 400ft^2
diagonal^2 = 800ft^2
diagonal = sqrt(800ft^2)

Now, we have the base (diagonal) and the height of the right triangle. We can find the length of the wire (hypotenuse) using the Pythagorean Theorem again:

length of wire^2 = base^2 + height^2
length of wire^2 = diagonal^2 + height^2
length of wire^2 = (sqrt(800ft^2))^2 + (16ft)^2
length of wire^2 = 800ft^2 + 256ft^2
length of wire^2 = 1056ft^2
length of wire = sqrt(1056ft^2)

Finally, we calculate the square root of 1056ft^2 to find the length of each wire:

length of wire ≈ 32.5ft

Therefore, the length of each wire should be approximately 32.5ft.