I don't see how
100-0.25x^2 =ax^2 - 80ax + 1600a
becomes
x^2 + (80a)/(-0.25-a)x + (1600a+100)/(-0.25-a) = 0
it doesn't ...
$\displaystyle 100-0.25x^2 =ax^2 - 80ax + 1600a$
$\displaystyle -ax^2 -0.25x^2 + 80ax - 1600a + 100 = 0$
$\displaystyle -(a + 0.25)x^2 + 80ax - 1600a + 100 = 0$
$\displaystyle x^2 + \dfrac{80ax}{-(a+0.25)} - \dfrac{1600a - 100}{-(a+0.25)} = 0$
$\displaystyle x^2 - \dfrac{80a}{a+0.25} \cdot x + \dfrac{1600a - 100}{a+0.25} = 0$
http://www.mathhelpforum.com/math-he...em-175040.html
sorry, we're not supposed to use calculus to solve the problem - i found an answer online and i'm trying to interpret it but i'm having some difficulty as u can see
ANSWER:
find the point, where curve 1 and curve 2 touch.
100-0.25x^2 = a(x-40^2)
100-0,25x^2 =ax^2 - 80ax + 40^2*a
...
x^2 + (80a)/(-0.25-a)*x + (-40^2a+100)/(-0.25-a) = 0
x = - (40a)/(-0.25-a) +- √(40^2a^2/(-0.25-a)^2 - (-40^2a+100)/(-0,25-a))
now to have only 1 tangent point, the term under the root must be = 0
40^2a^2/(-0,25-a)^2 = (-40^2a+100)/(-0,25-a)
solution:
a = 0,083333
now find x:
x = - 40*0,08333/(-0,25-0,08333)
x = 10 m
The curves touch at x = 10 m,
and the slope of curve 1 at 10 m is
y' = -1/2x = -5
and of curve 2 at 10 m is
y' = 0,08333(2x-80) = 0,08333*- 60 = - 5
i.e., the slopes are identical --> nice smooth ride
I understand what the answerer did but it's the intermediate steps that are confusing me - that is why I posted a part of the answer trying to figure out what the answerer did
$\displaystyle 100 - .25x^2 = a(x-40)^2$
$\displaystyle 400 - x^2 = 4a(x^2 - 80x + 1600)
$
$\displaystyle 400 - x^2 = 4ax^2 - 320ax + 6400a$
$\displaystyle 0 = (4a+1)x^2 - 320ax + (6400a - 400)$
$\displaystyle x = \dfrac{320a \pm \sqrt{(-320a)^2 - 4(4a+1)(6400a - 400)}}{2(4a+1)}
$
for a single solution, the discriminant = 0 ...
$\displaystyle (-320a)^2 - 4(4a+1)(6400a - 400) = 0
$
$\displaystyle 320^2 a^2 = 4(4a+1)(6400a - 400)$
$\displaystyle 320^2 a^2 = 1600(4a+1)(16a-1)$
$\displaystyle 64a^2 = 64a^2 + 12a - 1
$
$\displaystyle a = \dfrac{1}{12}$
now go back and determine that $\displaystyle x = 10$