Help, Calculating height of Trapezium

[mathsgeek]

Homework Statement

I know the base of the trapezium is 20km long. I also know the area of it has to be 1x10^5 m^3. I was just wondering how would you be able to calculate the height of the trapezium without knowing the length of the top line, i want to do it so the height is maximised at angles lower than 40 degrees.

I have managed to get to the equation of:
h=2A/a+b where a=base, b=top line

Can somebody please guide me on how i can get to want i need? Thanks

 

[LCKurtz]

You mean h = 2A/(a+b), the parentheses are needed. You need one of h or b to calculate the other.

Can you explain more precisely what you mean by "the height is maximised at angles lower than 40 degrees"?

 

[mathsgeek]

Well the slope/angles of the triangles (at either end) must be equal to or lower than 40 degrees. I don't need much help with that as of yet, once I am able to find the height with the known angle that should be fine because ill be able to do it with other angles then. So basically, the bottom length must be 20km long, and the slope of the triangles must not be greater than 40 degrees. Thanks

 

[LCKurtz]

I somehow think you haven't stated the whole problem. If you assume it is an isosceles trapezoid with angle φ at the bottom, from the geometry you can get the equation:

A = ah - h²cot(φ)

This determines implicitly a relation between h and φ. Put in your numbers for A and a and find the max of h remembering that:

0 ≤ φ ≤ 2π/9

 

[mathsgeek]

Yeh i forgot to put that up, i got to the equation you showed but didnt know if it was correct. The thing is do i go from there, sub in the values for A and a, as well as the angle and use the quadratic formula to determine the height, because i try this nd get VERY big values. How can i do this because shouldn't i only get one value for h for the equation to meet the desired area and angle range? Thanks

 

[mathsgeek]

Bump
(Sorry for this but i don't know what time it is over where everyone is (sleeping etc))

 

[mathsgeek]

Anybody?

 

[LCKurtz]

"20km long. I also know the area of it has to be 1x10^5 m^3."

Just out of curiosity, what about those units? Area would be in units², not units³. So you should have written m². But did you really mean m² and not km²? Have you tried drawing it to scale?

 

[mathsgeek]

The area is m^2, but its not km^2.

 

[mathsgeek]

Just use a different value for the area and bottom length as that might make it easier to explain then ill be able to do the rest by myself with my values.

 

[mathsgeek]

Bump. How are you supposed to do it when you got two variables in the equation, both h and the angle?

 

[mathsgeek]

Please delete thread, problem has been solved