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?

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 .

Similarly, the length OA is .

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

+

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

Now, f'(x)=

f'(x)=0  gives,

x=3 or -5

but the length cannot be negetive.

thus x=3.

u = -(6/7)w

step-by-step explanation:

place all values with 'u' on one side of the equation.

4u + 8w = - 3u + 2w

4u + 3u = 2w - 8w

further simplify the equation by solving the addition and subtraction.

7u = - 6w

make 'u' the independent variable by placing its co-efficient on the right-hand side.

u = - (6/7)w