The length of string is 54 cm

Question

A piece of string whose length is given below is cut into two pieces. One piece is used to form an equilateral triangle and the other to form a circle. What should be the perimeter of the equilateral triangle and the circumference of the circle so that the sum of the areas is a minimum? Find the minimum sum of the areas. Express your answer in terms of

Answer 1

Let the length of a side of the triangle = S and the diameter of the circle = D. Thus 3S+πD=54. Let A be the sum of the area of the triangle and the circle thus .5S²+πD²/4=A. Find dA/dS=S+πD/2dD/dS. Note 3+πdD/dS=0, or DD/dS=-3/π. Set dA/dS=0=S+3-3πD/2 to find S=3πD/2-3 for minimum area. Substitute this value of S to find 9πD/2-9+πD=54 and solve for D=7.07 and find S=10.6. Use those numbers to find A= 95.4 cm²

Answer 2

I will not work this problem for you, but I will tell you how to do it.

3x+2πr=54 where x is the side of the equilateral triangle and r the radius of the circle.

The sum of the areas is A= [x sqrt(3)/2] + πr².

Now use the first equation to express x in terms of r and substitute in the equation for A.

Then differentiate A with respect to r and set the derivative equal to 0.

Solve for r and back substitute for x.

This will be messy and require careful algebraic manipulation!