Roller Coaster Polynomial that describes height of ride at given distances

[iv7]

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)

 

[mmm4444bot]

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)

What have you done, so far? Where did you get stuck?

Please read the forum guidelines. Thank you! :cool:

 

[iv7]

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

 

[iv7]

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?

 

[mmm4444bot]

I came up with ax(x-20)(x-25)(x-50) and [found] a = -0.0025 …

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]

Does that mean it would be (x-50)^2 ?

 

[mmm4444bot]

Does that mean it would be (x-50)^2 ?

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]

I got 0.00000625 as a. is that correct?

 

[mmm4444bot]

I got 0.00000625 as a. is that correct?

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}