Finding the right length to cut a wire for 2 shapes

[wduk]

Hello

I am stuck on a problem i don't know how to solve. I have a wire of a length of 10cm i have to cut into two parts. I have to make a circle and a equilateral triangle.

So i had to express radius r in terms of length L so:

2pi*r +3L is my sum for total perimeters then to express r i have r = 3L/2pi

I then calculated the areas where triangle area is (sqrt(3)*L²) / 4 and my circle is pi(3L/2pi)²:

So total area a:
a = (sqrt(3)*L²)/4 + pi(3L/2pi)²

Now i get stuck as i have to find the value of L for which a takes a minimum/maximum value.

So i differentiated a to get:

L(pi*sqrt(3)+9)/2pi

Then i was trying to find zeros for this but the only value would be actually L = 0 so i am assuming i got this part wrong? Or maybe even an earlier part of it.

This is confusing me a lot hope you can help.

 

[ksdhart2]

Okay, so I'm assuming that an unstated part of the instructions was to maximize the area of the circle and the triangle. Without that bit of information, there's no one "correct" answer, as there are infinitely many ways to use a length of wire to form both a circle and an equilateral triangle. Assuming that's a proper interpretation of the problem statement, you're mostly on the right track, but you've made a simple arithmetic error when you solved for r. Everything's correct up until this point:

\(\displaystyle 2\pi \cdot r + 3L = 10\)

where r is the radius of the circle, and L is the length of one side of the equilateral triangle. The error in the next step came because you forgot that the above expression was equal to 10. Solving for r leaves:

\(\displaystyle r= \dfrac{10 - 3L}{2\pi}\)

Then we can use formulas for area of a circle and an equilateral triangle respectively and find that the sum of both areas is

\(\displaystyle A(L) = \pi \cdot \left( \dfrac{10 - 3L}{2\pi} \right)^2 + \dfrac{\sqrt{3}}{4} \cdot L^2\)

Try differentiating that and set it equal to 0. That should give you the right answer(s).

 

[MarkFL]

I would likely use optimization with constraint here, i.e. Lagrange Multipliers.

Let's let:

\(\displaystyle 0<W\) = the length of wire.

\(\displaystyle 0\le C\) = the length of wire used to form the circle.

\(\displaystyle 0\le T\) = the length of wire used to form the equilateral triangle.

Now, the objective function is:

\(\displaystyle \displaystyle f(C,T)=\frac{C^2}{4\pi}+\frac{1}{4\sqrt{3}}T^2\)

Subject to the constraint:

\(\displaystyle \displaystyle g(C,T)=C+T-W=0\)

And so we obtain the system:

\(\displaystyle \displaystyle \frac{C}{2\pi}=\lambda\)

\(\displaystyle \displaystyle \frac{T}{2\sqrt{3}}=\lambda\)

What does this imply?