Integration question... or is this a conics question? :s?

Question

"Design a semi-elliptical bridge big enough for 2 semi-trailers to fit under at one time with a minimum 1.5m separation between them and 0.5m clearance each side. Semi trailers are 2.8m and 4.6m high"

I'm thinking maybe not a conics question because a formula and the focus point wasn't given. Help!!!

Answer 1

I think the trailer length should be 4.6 m and its height 2.8 m right? You have it the other way around.
So two trailers with the required spacing have a combined length of 11.7 m (2*4.6 + 1.5 + 2*0.5) with a height of 3.3 m (2.8 + 0.5).
These are NOT the dimension of the elipse parameters, but rather the dimensions of a rectangle than can be inscribed inside the elipse.

So let hte center of the elipse be (0,0). Then a point on the elipse has cordinates (5.85, 3.3), where 5.85 = 11.3 / 2.
So we have the equation of the elipse
(x/a)^2 + (y/b)^2 = 1
where a and b are related by
(5.85/a)^2 + (3.3/b)^2 = 1

You could choose any values of a and b which fit that equation.

If you wish to choose a and b so as to minimize the circumference of the semiellipse, then you can use the following relation:
C =~ π*sqrt((a2+b2)/2)
I'm getting b = 8.03 and a = 6.42.

So equation is: (x/6.42)^2 + (y/8.03)^2 = 1

Answer 2

It's a conics question

so the general formula for an ellipse is
(x-h)^2/a^2 + (y-k)^2/b^2 = 1

leave out the h and the k's and assume that the ellipse is centered around (0,0).

Now, imagine a rectangle which symbolizes the semi's and the clearance. This rectangle is 2*2.8 + 1.5 + 1 m's wide, and 5.1 tall (assuming .5 meter clearance in all direction). This rectangle then touches the ellipse, and therefore you know that the ellipse must pass through (3.55, 5.1)

so then you have
3.55^2/a^2 + 5.1^2/b^2 = 1

pick an A less than 3.55, and then solve for be. There are an infinite number of ellipses that satisfy these conditions.

Answer 3

You do not say how wide the semi-trailers are. You do not indicate if the two semi-trailers must both be 4.6 m high. Is this a four lane highway that is running under the bridge? Is the 4.6 M high trailer always in the inside lane and the 2.8m trailer always in the outside (curb) lane?

Please rewrite the question clarifying these questions. Then a solution can be easily generated.

Answer 4

A cone has a million/3 the quantity of the cylinder it suits interior. using comparable triangles you are able to discern that the peak H of the finished cone that this would properly be a element of could be H = rh/(r - a) the quantity of the vast cone then could be V1 = a million/3 ?r²H = a million/3 ?r³h/(r - a) the peak of the lacking piece of the cone is H - h = ah/(r - a), and so its volume is V2 = a million/3 ?a²(H - h) = a million/3 ?a³h/(r - a) Then the quantity of the section is the vast volume minus the infant: V = a million/3?(r³ - a³)h/(r - a) using that r³ - a³ = (r - a)(r² + ra + a²), this would additionally be written as: V = a million/3?(r² + ra + a²)h