minimum wire length between points help please?

Question

This is a strange one, there is a tower 108 feet tall, and another 75 feet tall they are 270 feet apart, if a wire is strung from the top of the tower a to to the ground and then up to tower b, what is the minimum length of wire, and how far away from each tower would the point be where the wire touches the ground.

Answer 1

i dont know if you know calculus yet, but its about finding a minimum amount of wire. So first find the equation of the length of wire needed. call x a point between the buildings, and y the length of wire. The length for the first building = (x^2 - 108^2)^.5 (pythag the two lengths of the triangle). For the second building, its ((270-x)^2 + 75^2)^.5, pythag again but this time the bottom length = 270 - x. So y = (x^2-108^2)^.5 + ((270-x)^2 - 75^2)^.5
now you'll have to use calculus to find the minimum point on this equation. (or you could graph it and get your calculator to find the min for you)
y'= x/(x^2-108^2)^.5 + (x-270)/((270-x)^2 - 75^2)^.5
if you solve that for y' = 0 youll get it. You can do it the long by finding the comon denominator of both fractions, then adding the fractions together, and then solving for whatever makes the top of the fraction = 0, but that would take to long to show on a computer. Alternatively you could jsut throw it into your calculaotr

Answer 2

The wire runs from the top of the tall tower to a point on the ground between the two towers then back up to the top of the short tower.

If we use the ground as a mirror and run the wire from the top of the tall tower thru the ground to the top of the reflection of the short tower (below the ground) it will form a straight line which is the shortest distance. The length of the reflection part of the line (under the ground) is the same as the actual length back up.

So we have:

v = vertical distance = 108 + 75 = 183
h = horizontal distance = 270
T = total distance

T = √(v² + h²) = √(183² + 270²) = √(33,489 + 72,900)
T = √106,389 ≈ 326.17327 ft

We have two similar right triangles. The large one is described above. The small one is:

height = 108
horizontal = (108/183)(270) = 159.34426 ft

So the point where the wire touched the ground between the two towers was

159.34426 feet from the base of the tall tower and

270 - 159.34426 = 110.65574 feet from the base of the short tower

Answer 3

<>I'm way out of touch here, but I ran a proportion using 75/108=x/270 to get the ground lengths to be 187.5' and 82.5'. Then I use the Pythagorean Theorem to solve the 2 hypotenuses of the 2 right triangles formed and got 216.38' and 111.5' for the 2 lengths of wire (327.88' total wire length). Don't know if that's even close, but it's the best I could come up with with my 40 y/o math skills! :)

Answer 4

minimum length of wire will be obtaind if you peg the wire to the ground forming two straight lines of wire. You need to set up an equation for length or wire based on point it touches the ground (x). Differential wrt x, to find minimum length. you'll need some trig too!

good luckl ... sounds like fun!

Answer 5

enable L be the full length. L = sqrt(12^2+x^2) + sqrt[20^2+(10-x)^2] Differentiate w.r.t. x, L' = x/sqrt(12^2+x^2) + 2(10-x)/sqrt[20^2+(10-x)^2] = 0 are you able to end it now?