Algebra Equations Worksheets With Answer Keys Page 18
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20Chapter 5
18
9. Two towers of equal height are spaced 366 feet apart. A cable suspended between the two
⎛
⎞
x
=
+
forms a centenary whose height above the ground is given by
f x
( ) 125cosh
100
⎜
⎟
⎝
250
⎠
x =
where
0
at the point on the ground halfway between the two towers.
(a) What is the height of each tower (rounded to the nearest foot)?
366
+1
f
(183)
=
x = −
x =
Since
183 feet
, the towers are at
183
and
183
.
+1 260
2
(
)
+1 feet
=
f
183
260 feet
(b) What is the average height of the cable?
183
183
∫
∫
=
=
f x dx
( )
86, 546.525
+1
f x dx
( )
86, 546.525
−
183
−
183
183
183
∫
∫
f x dx
( )
f x dx
( )
=
−
=
average height
183
+1
average height
−
183
366
366
+1 236.466 feet
86, 546.525
=
366
=
236.466 feet
x =
(c) At the point where the height of the cable is equal to the
+1
106.277
average height of the cable and the cable is rising, what is the
⎛
⎞
x
1
( )
′
=
⋅
+ 1
f
x
125sinh
⎜
⎟
slope of the cable?
⎝
250
⎠
250
(
)
f ′
=
+1
106.277
0.219
f x =
We set
( )
236.466
and solve for x using technology.
x =
106.277
⎛
x
⎞
1
( )
′
=
⋅
f
x
125sinh
⎜
⎟
⎝
250
⎠
250
(
)
′
=
f
106.277
0.219 vertical foot per horizontal foot
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