Question

Two vertical poles of heights 60 ft and 80 ft stand on level ground, with their bases 100 ft apart. A cable that is stretched from the top of one pole to some point on the ground between the poles, and then to the top of the other pole. What is the minimum possible length of cable required

Answer 1

Answer:

Step-by-step explanation:

Given

height of two Poles are 60 and 80 ft

Distance between them is 100 ft

Let x be the distance of Pole of ht 80 ft from Point of stretch

thus length of rope is given by

$L=L_1+L_2$

$L_1=\sqrt{80^2+x^2}$

$L_2=\sqrt{60^2+(100-x)^2}$

$L=\sqrt{80^2+x^2}+\sqrt{60^2+(100-x)^2}$

differentiate w.r.t x we get

$\frac{\mathrm{d} L}{\mathrm{d} x}=\frac{2x}{2\sqrt{80^2+x^2}}-\frac{2\left ( 100-x\right )}{\sqrt{60^2+\left ( 100-2x\right )^2}}$

Put $\frac{\mathrm{d} L}{\mathrm{d} x}=0$ to get minimum value

$\frac{2x}{2\sqrt{80^2+x^2}}=\frac{2\left ( 100-x\right )}{\sqrt{60^2+\left ( 100-2x\right )^2}}$

squaring

$x^2\left ( (100-x)^2+60^2\right )=(100-x)^2(80^2+x^2)$

Rearranging

$28x^2-1800x+640000=0$

$x=\frac{400}{7}$

thus $L_1=98.312 ft$

$L_2=73.73 ft$

$L=172.04 ft$  

Answer 2

By using right triangles such that the cable is the hypotenuse of the triangles, we will see that the minimum length is:

L =  172.55 ft

How to get the minimum length of cable?

The minimum length is the length that we get when the cable forms the hypotenuses of two right triangles, such that the cathetus are the height of the poles and half of the distance between their bases.

So, for one of the triangles the cathetus measure 80ft and 50 ft (half of the distance between the two poles), then if we use the Pythagorean theorem, the length of the hypotenuse is:

H^2 = 80ft^2 + 50ft^2

H = √(80ft^2 + 50ft^2 ) = 94.45 ft

For the other pole, the cathetus measures 60ft and 50ft, then the hypotenuse in this case is:

H^2 = 60ft^2 + 50ft^2

H = √(60ft^2 + 50ft^2 ) = 78.10 ft

The total length of the cable will be:

L = 94.45 ft + 78.10 ft = 172.55 ft

If you want to learn more about right triangles, you can read:

https://brainly.com/question/2217700