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Two vertical poles, one 4 ft high and the other 16 ft high, stand 15 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?

Respuesta :

Answer:

The rope should be staked at 3ft  distance from bottom of  4 ft pole.

Step-by-step explanation:

Given that  the poles are 4 ft and 16 ft, separated by distance of 15 ft.

Refering the given figure, let the distance from 4 ft pole be "x".

consequently, distance from 16 ft pole is (15-x).

Refering the figure, in right angled triangle DCO, by pythagoras theorm, the the length OD is [tex]\sqrt{16-x^{2} }[/tex].

Similarly, the length OA is [tex]\sqrt{481+x^{2}-30x}[/tex].

thus the length of rope is AO + OD =  L = f(x) =

[tex]\sqrt{16-x^{2} }[/tex] + [tex]\sqrt{481+x^{2}-30x}[/tex]

now, for a function of one variable, to have maximum or minimum, differentiate once, and find value of x.

Now, f'(x)= [tex]x(\sqrt{16+x^{2} }^{-1} )+ (x-15)(\sqrt{481+x^{2}-30x }^{-1})[/tex]

f'(x)=0  gives,

x=3 or -5

but the length cannot be negetive.

thus x=3.

Ver imagen mohan67
akposevictor

By applying the pythagroean theorem and differentiation, the distance from the bottom of the pole that the rope should be staked at is: 3 ft.

What is the Pythagorean Theorem?

The pythagorean theorem states that the square of the hypotenuse of a right triangle equals the sum the square of the otehr two legs of the right triangle.

The sketch of the problem is given in the diagram that is attached below.

Applying the pythagorean theorem in right triangle DCO, we would have:

DO = √(16 - x²)

AO would also be found using the pythagorean theorem:

AO = √(481 + x² - 30x)

Therefore, the length of the rope = AO + DO = f(x), which is:

f(x) = √(481 + x² - 30x) + √(16 - x²)

To find the value of x, take derivative and set equal to 0.

Differentiating, f(x) = √(481 + x² - 30x) + √(16 - x²), we would have:

f'(x) = 0 which will give us, x = 3 or -5.

Since the length cannot be negative, therefore, the distance from the bottom of the pole that the rope should be staked at is: 3 ft.

Learn more about pythagorean theorem on:

https://brainly.com/question/654982

Ver imagen akposevictor