A wire is stretched from the ground to the top of an antenna tower. The wire is 20 ft. long. The height of the tower is 4 ft. greater than the distance d from the tower's base to the bottom of the wire. Find the distance d and the height of the tower.
2007-09-09 15:11:52 · 6 answers · asked by Brandon 1 in Science & Mathematics ➔ Mathematics
use Pyth. Th.
(d+4)^2 + d^2 = 20^2
(d+4)(d+4) + d^2 = 400
2d^2 + 8d + 16 = 400
d^2 + 4d - 192 = 0
(d+16)(d-12) = 0
d = -16 or 12, but only 12 makes sense
d = 12 ft and tower = 16 ft
2007-09-09 15:19:00 · answer #1 · answered by Jonathan S 2 · 0⤊ 0⤋
You are basically describing a right angled triangle (RAT) here.
With hypotenuse = 20 and other sides d and h
Two ways to solves this
1) Knowledge of RAT triplets:
(3, 4, 5) is a well known set of lengths for the sides of a RAT and any multiple of can also describe the sides of a RAT (e.g (6, 8, 10))
We have a hypotenuse of 20 = 4*5
We have been told that d and h (other 2 sides) differ by 4
So (12, 16, 20) would fit the criteria
Giving h=16 and d=12
2) Solving for d and h:
Pythag. theory : h^2 + d^2 = 20^2 = 400 (1)
Given that d = h-4 (2)
Put (2) into (1) and simplify with all terms on one side
h^2 + (h-4)^2 = 400
=>2h^2 - 8h -384 =0
=>h^2 - 4h -192 =0
Factorise
=>(h+12)(h-16)=0
=> h=-12 or h=16
Reject negative solution (can't have negative length)
h=16 => d=12 (same answer as found through knowledge of RAT triplets)
2007-09-09 22:33:48 · answer #2 · answered by piscesgirl 3 · 0⤊ 0⤋
use pythagarian theorem
let x be the distance from the ground to the base of the tower. then x + 4 is the height of th tower
a^2 + b^2 = c^2
x^2 + (x + 4)^2 = 20^2
x^2 + x^2 + 8x + 16 = 400
2x^2 + 8x - 384 = 0
use qudratic formula and you'll get x= 12ft
height: = 16ft
d = 12 ft
2007-09-09 22:21:00 · answer #3 · answered by 7 · 0⤊ 0⤋
You have a right angled triangle formed by the wire (20ft) the ground (d) and the tower (d + 4)
Using pythagoras
20² = d² + (d + 4)²
400 = 2d² + 8d + 16
200 = d² + 4d + 8
d² + 4d - 192 = 0
Using the quadratic formula
d = [-4 ± â4² - 4(-192)] /2
d = [-4 ± â16 + 768] /2
d = [-4 ± â784]/2
d = [-4 ± 28]/2
d must be positive
d = [-4 + 28]/2
d = 24/2
d = 12
d + 4 = 16
d = 12ft
tower = 16ft tall
2007-09-09 22:21:45 · answer #4 · answered by Tom :: Athier than Thou 6 · 0⤊ 0⤋
d^2 +(d+4)^2 = 20^2
d^2 +d^2 +8d + 16 = 400
2d^2 +8d +16 =400
d^2 +4d -192 = 0
(d+16)(d-12) =0
d = 12
d+4 = 16 = height of tower
2007-09-09 22:27:53 · answer #5 · answered by ironduke8159 7 · 0⤊ 0⤋
D^2+4^2+D^2=20^2
D^2+16+D^2=400
D^2+D^2=384
D+D=19.6
2D=19.6
D=9.8
2007-09-09 22:29:17 · answer #6 · answered by Paul K 1 · 0⤊ 2⤋