# A covered metal triangular trough is constructed as follows: A square shaped sheet

QuestionsCategory: CalculusA covered metal triangular trough is constructed as follows: A square shaped sheet
A covered metal triangular trough is constructed as follows:

A square shaped sheet of metal which is 70 centimeters wide and long square is folded along the center. Next, two pieces of metal in the shape of isosceles triangles are are welded to the ends. Finally, a metal cover is attached to the top).

We want to find the smallest and largest surfaces area S, that a so constructed trough can have, and at what opening angle w of the triangular pieces it is attained. Proceed as follows:

need to know:

Find the surfaces area as a function of the angle. (Be sure to include all five sides of the trough).
— S(w) = ?

— natural domain of S(w) (left and right end point)

On its domain S(w) has one stationary point, w=c. Although there is no explicit formula for the value of c, itself, it is possible to derive the exact value of the cosine of c,. Find it:
— cos(c) =

Find the surface area at the stationary point
— S(c) =

Find also the global minimal value of S in its natural domain

The surface area will be 70^2 + ends + top.

End surface area will be dependent on the angle:
altitude/35=sinTheta
area= 1/2 35*altitude= 1/2 *35^2*sinTheta
so two ends…35^2 sinTheta.
Now the top: 35*d where d is found by the law of sines: d/sinTheta=35/sin[1/2(180-theta)]
area top= 35*sintheat*35/sin(90-theta/2)

total area= sum of above.