Two vertical poles, one 16 ft high and the other 24 ft high, stand 30 feet apart on a flat field. A worker wants to support both poles by running rope from the ground to the top of each post. If the worker wants to stake both ropes in the ground at the same point, where should the stake be placed to use the least amount of rope?

The least amount of the rope is the smallest length that can be gotten from the pole

The rope should be placed at 12 feet from the 16 ft pole to use the least amount of rope.

The heights are given as:

The distance is given as:

See attachment for the illustrating diagram

Considering the two right-angled triangles on the diagram, we have the following equations, using Pythagoras theorem

Expand

The length (L) of the pole is:

So, we have:

Differentiate

Set to 0

Take LCM

Simplify

Rewrite as:

Square both sides

Expand

Simplify

Evaluate like terms

Rewrite as:

Divide through by 320

Using a calculator, we have:

The value of x cannot be negative.

So, we have:

Hence, the rope should be placed at 12 feet from the 16 ft pole

Read more about minimizing lengths at:

link

The rope should be placed at 1.46 from the 16 ft pole to minimize the length.

Step-by-step explanation:

From the diagram, our goal is to minimize the length of Rope AC passing through B.

First, we determine the length of the rope AC.

AC=AB+BC

In the first triangle,

Similarly, in the second triangle,

Length of the Rope, AC

First, to minimize L,we find its derivative.

Setting the derivative to zero

The rope should be placed at 1.46 from the 16 ft pole to minimize the length.

complete the steps to find the area of the trapezoid.

area of rectangle = 256   square units

area of triangle 1 = 24  square units

area of triangle 2 = 16  square units

area of triangle 3 =   96 square units

area of trapezoid = 120  square units

step-by-step explanation: