# Thread: Need to find the function of the graph

1. ## Need to find the function of the graph

The largest, longest and highest roller coaster in the world is the "Colossus" in Los Angeles. The longest straight section of "Colossus" goes through a horizontal distance of 330m and the shape of the roller coaster along this section may be represented by a 4th degree polynomial function.

At the start of this section of the "Colossus", its height off the ground is 150m (assuming we going from left-to-right). 60m Horizontally along, the "Colossus" has dipped to a height of just 30m. It peaks again at a horizontal distance of 180m and features another dip at a horizontal distance of 300m.

The question is:
Using metres as your units use calculus methods to find the function that describes the curve of this section of the "Colossus"

You need to use simultaneous equations.

2. Originally Posted by mortalcyrax
The largest, longest and highest roller coaster in the world is the "Colossus" in Los Angeles. The longest straight section of "Colossus" goes through a horizontal distance of 330m and the shape of the roller coaster along this section may be represented by a 4th degree polynomial function.

At the start of this section of the "Colossus", its height off the ground is 150m (assuming we going from left-to-right). 60m Horizontally along, the "Colossus" has dipped to a height of just 30m. It peaks again at a horizontal distance of 180m and features another dip at a horizontal distance of 300m.

The question is:
Using metres as your units use calculus methods to find the function that describes the curve of this section of the "Colossus"

You need to use simultaneous equations.

The problems says "may be represented by a 4th degree equation. That is $y= a+ bx+ cx^2+ dx^3+ ex^4$ where x is the horizontal distance and y the height. You need to find the 5 coefficients so you need 5 equations.

"At the start of this section of the "Colossus", its height off the ground is 150m " so y(0)= a= 150.

"60m Horizontally along, the "Colossus" has dipped to a height of just 30m" so $y(60)= 150+ 60b+ 60^2c+ 60^3d+ 60^4e= 30$.

From the way this is worded, I think we can assume this is a minimum height so the derivative must be 0: $b+ 2(60)c+ 3(60^2)d+ 4(60^3)e= 0$

"It peaks again at a horizontal distance of 180m". We aren't given the height but again the derivative must be 0: $b+ 2(180)c+ 3(180^2)d+ 4(180^3)e= 0$.

"and features another dip at a horizontal distance of 300m. " Again we aren't told the height but the derivative must be 0: $b+ 2(300)c+ 3(300^2)d+ 4(300^3)e= 0$.

Five eqations to solve for a, b, c, d, and e.

3. Originally Posted by HallsofIvy
The problems says "may be represented by a 4th degree equation. That is $y= a+ bx+ cx^2+ dx^3+ ex^4$ where x is the horizontal distance and y the height. You need to find the 5 coefficients so you need 5 equations.

"At the start of this section of the "Colossus", its height off the ground is 150m " so y(0)= a= 150.

"60m Horizontally along, the "Colossus" has dipped to a height of just 30m" so $y(60)= 150+ 60b+ 60^2c+ 60^3d+ 60^4e= 30$.

From the way this is worded, I think we can assume this is a minimum height so the derivative must be 0: $b+ 2(60)c+ 3(60^2)d+ 4(60^3)e= 0$

"It peaks again at a horizontal distance of 180m". We aren't given the height but again the derivative must be 0: [tex]b+ 2(180)c+ 3(180^2)d+ 4(180^3)e= 0.

"and features another dip at a horizontal distance of 300m. " Again we aren't told the height but the derivative must be 0: [tex]b+ 2(300)c+ 3(300^2)d+ 4(300^3)e= 0.

Five eqations to solve for a, b, c, d, and e.
Thank you but where do I go from here?

4. Solve the five equations for a, b, c, d, and e like I said!

5. Originally Posted by HallsofIvy
Solve the five equations for a, b, c, d, and e like I said!
I know but I don't know how to do that. Could you help me do one of the equations if possible?

6. No, because you cannot do one equation. If you honestly don't know how to solve simultaneous linear equations then you shouldn't be trying a problem like this.