How do i find maxiumum volume of a triangular prism?

Question

I have to find the volume of a triangular prism.
It says: farmer pete wants to build a cattle trough using supplies he already has, but wants it to hold the most feed possible. The trough is 4meters long and its cross sections are all isosceles triangles with 2 equal sides of 40cm in length. The angle between the two equal sides is θ (theta).

1. refresh your memory about area of triangles and volume of triangular prisms. List the triangle are and formula in your final write-up as well as the volume formula. Define variables, explain using a picture, how they "fit" into this situation.

2. Find the area of the "base", the triable of the prism, as a function of θ/2. A(θ/2)=_____

3. Convert your Area function: A(θ/2), to a function in terms of θ. That is: A(θ)=_____ (recall the double angle formulas)

Answer 1

Hi Lora,

 

There is actually an easier method for solving this problem using the fact that the area if a triangle when you know two sides (a and b) and an included angle is given by (absinΘ)/2.

 

Sounds like the intent of this problem is to not use this formula, but rather to derive the formula for the area by dividing the isosceles triangle into two right triangles the using sin and cos ratios to get the base and height of the original triangle so you can find its area. Of course the volume of the prism is the area of the base times its height.

 

So drop a perpendicular from the vertex angle to the base of the triangle. This will bisect the vertex angle into two angles each of measure theta/2. The sin theta/2 = x/40 where x represents 1/2 the base of the original isosceles triangle.

 

So x = 40 sin theta/2 and the base of the isos triangle is 80 sin theta/2

 

The height of the isos triangle is found by cos theta/2 = h/40. So h = 40 cos theta/2

 

The area of the triangle is then A = 1/2 (80 sin theta/2) (40 cos theta/2) = 1600 sin theta/2 cos theta/2.

 

Now here is where the double angle formula comes into play: Sin 2theta = 2 sin theta cos theta, so we can rewrite area as A = 1600/2 (2 sin theta/2 cos theta/2) = 800 sin theta.

 

V = 3200 sin theta

 

V becomes a maximum when theta is 90 degrees (sin 90 = 1). So the max volume is 3200 cubic m.

 

It will help to sketch the isosceles triangle, drop the perpendicular to the base, show theta/2 at the vertex angle and label x, b, h.