• Mar 19th 2011, 04:58 AM
LandLBoy
The descent of a roller coaster starts at the max point of a parabola whose equation is y= 100 - 1/4x^2 and ends along the horizontal line y = 0. In order to make the ride smooth, a section of track following the graph of the function y = a(x-40)^2 is added as a transition from the parabola to the line.

a) determine the value for ( a ) which makes the ride from the top to the end along the horizontal line smooth.

b) At what point do the two curves meet?

c) What is the slope of the tangent at this point?

I know I should probably some effort but this question is totally stumping me and I have no idea where to start...Can someone at least set me on the right track?
• Mar 19th 2011, 06:15 AM
skeeter
Quote:

Originally Posted by LandLBoy
The descent of a roller coaster starts at the max point of a parabola whose equation is y= 100 - 1/4x^2 and ends along the horizontal line y = 0. In order to make the ride smooth, a section of track following the graph of the function y = a(x-40)^2 is added as a transition from the parabola to the line.

a) determine the value for ( a ) which makes the ride from the top to the end along the horizontal line smooth.

b) At what point do the two curves meet?

c) What is the slope of the tangent at this point?

I know I should probably some effort but this question is totally stumping me and I have no idea where to start...Can someone at least set me on the right track?

curves are equal at some value $\displaystyle 0 < x < 40$ ...

$\displaystyle 100 - \dfrac{x^2}{4} = a(x-40)^2$

slopes are equal at the same value $\displaystyle 0 < x < 40$ ...

$\displaystyle -\frac{x}{2} = 2a(x-40)$

solving for $\displaystyle a$ from the derivative equation ...

$\displaystyle a = -\dfrac{x}{4(x-40)}$

sub in for $\displaystyle a$ in the first equation and solve for $\displaystyle x$ and $\displaystyle a$
• Mar 19th 2011, 06:49 AM
NOX Andrew