Calculus question (help me plz)

[Guest]

Hey I really need help with a homework question my teacher gave me. It goes like this:- A car is travelling, at night, along a parabola shaped road. Its starting point is (100,100), its turning point is (0,0) and it's ending point at (-100,100). Besides the road there is a wall, it's base is at
(-100,50) and heading in a northerly direction for 30m, ending at (-100,80). If the car is travelling at 60km/hr, for how long will it's headlight illuminate the wall? Solve using Calculus Methods.

I just kind of need instructions on what to do and I can take it from there. So far I've drawn a graph of it and got a formula, x^2/100, but I'm not sure if it's right. Any help would be greatly appreciated.

 

[stapel]

A similar exercise was posted in this related thread. The reply suggested the following:

To find the equation of the parabolic path y = a(x - h)2 + k, note that (h, k) = (0, 0), and the point (-100, 100) lies on the curve.

Then note that the path of the headlights when the car is at any particular point (assuming the headlights are aimed straight ahead -- lawful adjustment would likely be too difficult to work with) will correspond to the tangent line at that point. The slope of that tangent line will be the derivative of the path's equation.

You have found the equation. Please see what sort of progress you can make with the second paragraph above.

If you get stuck, please reply showing what you have tried. Thank you.

Eliz.

 

[galactus]

Your parabola equation is correct. You'll need the derivative.

You need to find the equations of the two lines which are tangent to the parabola and pass through (-100,50) and (-100,80). Well, you at least need the coordinates of the points where they are tangent to the parabola.

This can be tricky because we don't know where the lines are tangent to the parabola.

We can use the point-slope form by subbing in the appropriate expressions for y, m, x.

y-y₁=m(x-x₁)

y=x²/100, m=x/50, y₁=50, x₁=-100.

((x²/100)-50=(x/50)(x+100), solve for x. Sub into parabola equation to get y.
This is the x and y value at the point where the line is tangent to the parabola. When the car is at this tangent point it's headlights will begin to hit the wall at (-100,50). You can use this data to find the slope-intercept form of the line.

Do the same for the other line. If you done them correctly, they will be tangent to the parabola and pass through the appropriate points(beginning and end of wall coordinates).

Once you have the respective x values for the tangent points you can use the arc length formula to find the length the car will travel.

Here's a picture:

The green line is where the car's headlights begin to hit the wall and the red line is where they hit at the end of the wall.

 

[Guest]

Thanks

Thanks for the help with this problem guys, I'm going to go and see if I can figure it out. You have definitely made it easier for me. One more thing before I go, what is the arc length formula? We havn't learnt it yet and we are only supposed to use the things that we have learnt, know any alternatives??

 

[Guest]

Am I right

Hey Galactus
I tried what you told me with the y-y1=m(x-x1) to work out the points on the parabola where the tangents run through. Anyway I got (-40,16) for the (-100,80) point and (-25, 6.25) for the (-100,50) point. I'm not sure if i'm right with the (-40,16) point it doesnt seem right? Can anyone help me plz, It's confusing my head