Pre-Calc question

[Shaded]

Hey everyone. I'm pretty new hear and just dipping my feet into calculus. I have one, two part question that I am not totally stumped with as part of it I believe I have finished, but I am a bit baffled on the other part. I don't expect answers, but anything that can help me narrow things down would be appreciated.

Here's the problem I'm working on:

The old highway, US 165 has at one point an old tunnel over the road in the shape of a parabolic arch. The span is 120 feet at the roadway with a maximum height of 25 feet.

A. Find a function to describe the height of the parabolic arch as a function of the distance from the center of the arch.

B. AAA Trucking is transporting a rectangular load of 15 feet width and an overall height of 23 feet along US 165. Can the truck clear the tunnel safely? Justify your answer.


I believe I have the answer for the first part, but it seems a bit wrong.
One I know the parabola is going to open downwards as it is a tunnel so the leading coefficient is going to be negative.
If I graph it out on a graph between 0 and 120 with a height of 25, I can divide 120 by 2 to find the x coordinate for the vertex.
This gives me a vertex of (60,25)
Using this I can put this in vertex form f(x)=a(x-h)^2+k
This gives me f(x)=a(x-60)^2+25

To find a I plugged in the point (120,0) as it is a known point and substituted them in for x and f(x).
After solving the equation I ended up with a 0.26 which gave me the final equation of f(x)=-0.26(x-60)^2+25.
After doing the math it seems right, but it doesn't look right on a calculator when I plug in the equation.
Any feedback I can get on this would be appreciated.

As for the second part I am drawing blanks on. Presuming my initial function is correct when f(x)=23 than X= 57.23 and 62.77 respectively. Because the difference of the two x values is less than 15 (the width) I want to say that the truck won't fit, but a few people in class who worked it out said it would fit, just barely. Am I working this out wrong? For some reason I don't feel like I am approaching the second part correctly.

Any help in this regard would be very much appreciated. I'm looking to understand how this is worked out more than anything.

 

[stapel]

Thank you for showing your work and reasoning so nicely! :cool:

The old highway, US 165 has at one point an old tunnel over the road in the shape of a parabolic arch. The span is 120 feet at the roadway with a maximum height of 25 feet.

A. Find a function to describe the height of the parabolic arch as a function of the distance from the center of the arch.

Simplify your computations by centering the parabola on the y-axis. Since the width is 120, then the x-intercepts will be at x = -60 and x = +60. The maximum height will be at (x, y) = (0, 25).

. . . . .y = a(x - 0)^2 + 25

Plug in one of the x-intercepts:

. . . . .0 = a(60 - 0)^2 + 25

. . . . .-25 = a(60)^2

. . . . .-25/60^2 = a

...which is negative, as expected.

. . . . .y = (-1/144)x^2 + 25

B. AAA Trucking is transporting a rectangular load of 15 feet width and an overall height of 23 feet along US 165. Can the truck clear the tunnel safely? Justify your answer.

If the width is 15, then the upper corners (assuming the truck drives smack down the middle in order to optimize "headroom") will be at x = -7.5 and x = +7.5. So you need to figure out if the height, y, at x = 7.5 is greater than 23. To find out, plug 7.5 in for x, and simplify to find y.

. . . . .y = (-1/144)(7.5)^2 + 25 = ...?

 

[Ishuda]

...To find a I plugged in the point (120,0) as it is a known point and substituted them in for x and f(x).
After solving the equation I ended up with a 0.26 which gave me the final equation of f(x)=-0.26(x-60)^2+25.
After doing the math it seems right, but it doesn't look right on a calculator when I plug in the equation.
Any feedback I can get on this would be appreciated.
...

Note that the 1/144 from stapel (which I agree with if it means anything) is not equal to 0.26. It should as it is the coefficient of x² and so I believe you have the right form but the wrong particular solution.

BTW: If you had the proper a, you could get the same answer using your equation if you put in x=60-7.5=52.5 (or 67.5) for your equation.

 

[Shaded]

Thanks!

Wow, I wasn't expecting a response, not to mention one so quickly. Once you suggested to center the tunnel on the y-axis things started to make a lot more sense and the answers came pretty quickly after. I do appreciate you working everything out for me though giving me something to verify my work with. Thanks a million!

I also started to see where I went wrong originally. I think I was probably just a little to eager to get out of class (it's friday lol) and tried to make more of it than it actually was.