Two posts, one 12 feet high and the other 28 feet high, stand 30 feet apart. They are to be stayed by two wires, attached to single stake, running from ground level to the top of each post. Where should the stake be placed to use the least wire?

If we place the origin (0,0) at the base of the 12' stake, then the 28' stake has its base at (30,0).

If the length of wire used is

L = sqrt(x^2 + 12^2) + sqrt((x-30)^2 + 28^2)
= sqrt(x^2 + 144) + sqrt(x^2 - 60x + 1684)

dL/dx = x/(sqrt(x^2 + 144) + (x-30)/sqrt(x^2 - 60x + 1684)

dL/dx = 0 when x=9

so, place the stake between the poles, 9' from the short pole, and 21' from the tall pole.

To find the optimal placement of the stake, we need to minimize the length of wire needed. Let's denote the distance between the stake and the 12-foot post as x, and the distance between the stake and the 28-foot post as 30 - x (since they are 30 feet apart).

Now, we can use the Pythagorean theorem to determine the length of each wire. The sum of the squares of the two lengths should be minimized.

The length of the wire attached to the 12-foot post can be calculated using the Pythagorean theorem:

Wire(length) = √(12^2 + x^2)

The length of the wire attached to the 28-foot post can be calculated using the Pythagorean theorem:

Wire(length) = √(28^2 + (30 - x)^2)

Now, to minimize the wire length, we need to find the derivative of the total wire length equation and set it equal to zero. Then we can solve for x.

Let's calculate the derivative and solve for x.

First, differentiate the equation of the total wire length with respect to x:

d(Wire(length))/dx = (1/2) * (12^2 + x^2)^(-1/2) * 2x + (1/2) * (28^2 + (30 - x)^2)^(-1/2) * (-2(30 - x))

Now set the derivative equal to zero and solve for x:

(1/2) * (12^2 + x^2)^(-1/2) * 2x + (1/2) * (28^2 + (30 - x)^2)^(-1/2) * (-2(30 - x)) = 0

Simplify the equation:

x / √(12^2 + x^2) - (30 - x) / √(28^2 + (30 - x)^2) = 0

Now, solve this equation to find the value of x.

To determine the position of the stake that uses the least amount of wire, we need to use the concept of similar triangles. Let's break down the problem.

We have two posts, one 12 feet high and the other 28 feet high, that are 30 feet apart. The wires will run from the ground to the top of each post and will be attached to a single stake.

Let's assume the stake is placed at a distance of x feet from the shorter post.

Using similar triangles, we can set up an equation to solve for x.

The ratio of the height of the shorter post (12 feet) to the distance from the stake to the shorter post (x feet) is equal to the ratio of the height of the taller post (28 feet) to the distance from the stake to the taller post (x + 30 feet).

12 / x = 28 / (x + 30)

To solve this equation, we can cross-multiply:

12(x + 30) = 28x

Expanding the equation:

12x + 360 = 28x

Rearranging the terms:

360 = 28x - 12x

360 = 16x

Now, divide both sides of the equation by 16:

x = 360 / 16

Simplifying:

x = 22.5

Therefore, the stake should be placed 22.5 feet from the shorter post to use the least amount of wire.