Problema Solution
The arch of a bridge is a parabola and six vertical cables that help support the road are equally spaced at 4m. The parabolic arch is in an x-y coordinate system, with the left-end of the arch at the origin. The length of the left most cable is 3.072m.
Find the (x-h)^2 = -4a(y-k) equation. What are the lengths of the other cables?
Answer provided by our tutors
Draw a diagram according to the text.
The parabola passes through (0,0).
There are 6 cables, 4 meters apart so the bridge is 4×7=28 metres long and hence the parabola passes through (28,0).
The first cable is 3.072 metres long and hence the parabola passes through (4,3.072).
We will substitute (0,0),(28,0) and (4,3.072) into the equation of the parabolas, (x−h)^2=−4a(y−k) and solve the resulting equations for h, k and a
(0−h)^2=−4a(0−k)
h^2 = 4ak
(28−h)^2=−4a(0−k)
(28−h)^2= 4ak
if we put h^2 on the place of 4ak we get the quadratic equation
(28−h)^2= h^2 and by solving it we find
h = 14
click here to see the step by step solution of the quadratic equations
now we have
4ak = 14^2
ak = 49
and
(4−14)^2=−4a(3.072−k)
−4a(3.072−k) = 100
by solving the system of equations
ak = 49
−4a(3.072−k) = 100
we find
a = 7.8125
k = 6.272
click here to see the step by step solution of the system of equations
thus the equation is
(x-14)^2 = -4*7.8125(y-6.272)
to find the lengths of the other cables we need to substitute the values for x and calculate y using the above equations that is
(x-14)^2 = -4*7.8125(y-6.272)
y = 6.272 - ((x-14)^2)/31.25
for x = 4 m
y = 6.272 - ((4-14)^2)/31.25
y = 3.072m
for x = 8m
y = 6.272 - ((8-14)^2)/31.25
y = 5.12m
for x = 12m
y = 6.272 - ((12-14)^2)/31.25
y = 6.144m
for x = 16m
y = 6.272 - ((16-14)^2)/31.25
y = 6.144m
x = 20m
y = 6.272 - ((20-14)^2)/31.25
y = 5.12m
for x = 24 m
y = 6.272 - ((24-14)^2)/31.25
y = 3.072m