Roller Coaster Polynomial that describes height of ride at given distances

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iv7

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You want to design a roller coaster. It will start at the ground level. You want the coaster to be at the ground level 20 yards away, 25 yards and 50 yards away. 10 yards away, you want the coaster to be at a height of 15 yards.

You decide to model the roller coaster with a polynomial where the coaster will be below the ground level between 20 and 25 yards. At 50 yards, the coaster will just "touch" the ground (i.e. the polynomial model will not "cross" the ground at 50 yards).

Create a polynomial model for the rollercoaster and graph it with a calculator. (Hint: the leading coefficient will be a very small number)
 
iv7 said:
You want to design a roller coaster. It will start at the ground level. You want the coaster to be at the ground level 20 yards away, 25 yards and 50 yards away. 10 yards away, you want the coaster to be at a height of 15 yards.

You decide to model the roller coaster with a polynomial where the coaster will be below the ground level between 20 and 25 yards. At 50 yards, the coaster will just "touch" the ground (i.e. the polynomial model will not "cross" the ground at 50 yards).

Create a polynomial model for the rollercoaster and graph it with a calculator. (Hint: the leading coefficient will be a very small number)
Click to expand...
What have you done, so far? Where did you get stuck?

Please read the forum guidelines. Thank you! :cool:
 
I came up with ax(x-20)(x-25)(x-50) and plugged in 10, 15 to get a = -0.0025 but it was wrong
 
roller coaster polynomial problem

You want to design a roller coaster. It will start at the ground level. You want the coaster to be at the ground level 20 yards away, 25 yards and 50 yards away. 10 yards away, you want the coaster to be at a height of 15 yards.

You decide to model the roller coaster with a polynomial where the coaster will be below the ground level between 20 and 25 yards. At 50 yards, the coaster will just "touch" the ground (i.e. the polynomial model will not "cross" the ground at 50 yards).

Create a polynomial model for the rollercoaster and graph it with a calculator. (Hint: the leading coefficient will be a very small number)

I got ax(x-20)(x-25)(x-50) and pulugged in 10, 15 to get a = -0.00025, but that was wrong. Any help as to why?
 
iv7 said:
I came up with ax(x-20)(x-25)(x-50) and [found] a = -0.0025 …
Click to expand...
That ought to be -0.00025. But it doesn't matter because the factor (x-50) is not correct. (Fixing the factorization will require recalculating a.)

We're told that the graph cannot cross the axis at x=50; it has to "just touch".

Look at the graphs for each of the following, as x goes from -5 to 5. What do you notice about the factors -- compared to the points where each graph touches the axis without crossing it?

y = -1/2*(x - 1)^2*(x + 2)

y = 1/10*(x - 4)*(x + 3)^2

Does this give you any idea about a simple change that you could make to your factorization below?

y = a*x*(x - 20)*(x - 25)*(x - 50)
 
iv7 said:
Does that mean it would be (x-50)^2 ?
Click to expand...
Yes! Repeated roots cause a polynomial's graph to touch the x-axis without crossing it.

What's your new value for the leading coefficient a?
 
iv7 said:
I got 0.00000625 as a. is that correct?
Click to expand...
That's what I got. Did you graph it, to check? :cool:

f(x) = 1/160000·x^5 - 29/32000·x^4 + 3/64·x^3 - 65/64·x^2 + 125/16·x


CLICK THUMBNAIL FOR LARGER IMAGE

coast50yd.JPG
 
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