Maths trig help please...?

Question

A farmer wishes to construct a paddock. Note a channel forms one side of the triangular paddock and he makes the other sides of the paddock using 225metres of available wire. These 2 sides need not be the same length but may be.

The problem is that the farmer wants this paddock to be the largest possible paddock using just 225m of wire or a paddock of the largest possible area.

Find the 2 wire lengths of the triangular paddock given:

a) the angle produced by the 2 sides is 90 degrees?

b) '' 70 degrees?

c) '' 120 degrees?

d) the angle can be of any magnitude or find not only the maximum or optimum side lengths and the best angle use?

Could you please explain how I would complete this as I am unsure.. I do not expect people to figure it out for me.

Thank you

Answer 1

a. Use c^2 = a^2 + b^2 for a 90 degree angle

you know b = 225 - a

Area = (1/2)ab = (1/2)a(225 - a)

This is just a parabola

Area will be 0 when a = 0 or a = 225

And the max will be half way inbetween these values giving a = 112.5

b. Use the law of cosines:

c^2 = a^2 + b^2 - 2abcos(70)

b = 225 - a

area = (1/2)a*bsin(70) = (1/2)a(225 - a)sin(70)

c.I think you know what to do now.

d.I think you know that the length of the side is. Also think about what the area represents. A rectangle maybe? And what type of rectangle gives maximum area? Something like that.

Answer 2

It doesn't say anything about the length of the third side (the "channel") or if it is fixed.  Nevertheless I think the key here is that if the base is fixed, the area is maximized when the height is maximized.

In the case of the 90 degree angle, we know that this happens for the isosceles right triangle, and the dimensions and area are easy to compute.  I believe that it is true for all the questions posed that the area will be maximized when the triangle is isosceles and both sides are an equal 112.5 m.