How Is Work Calculated When Stretching a String into an Equilateral Triangle?

[GregA]

Hi folks...can someone tell me where I'm going wrong with this question, or back me up if I'm right?

A string of natural length 2a and modulus of elasticity b has its ends attached to fixed points A and B where AB = 3a. Find the work done when the midpoint C of the string is pulled away from the line AB to a position where triangle ABC is equilateral. My attempt to solve is as follows:

I ignore B and focus just on A and C
if the natural length of the string = 2a then the midpoint represents half this length and therefore = a
AB = 3a, and so AC = 3a/2... the extention x = a/2
AC when streched to the equilateral triangle = 3a and so...x = 2a
The question asks me how much work is done from pulling the midpoint from its initial position on the line AB to its final position

Surely I want to subtract the work taken to pull the string to from its natural length to AC1 from the work required to pull the string to AC2..using Work = (b(x)^2)/2a...
(b((2a)^2-(a/2)^2)/2a = 15ab/8.
The books answer is given as 4ab however...I've tried to find a problem with my working and cannot.

 

[GregA]

I think I have it...There are two tensions opposing the movement of that string not just AC but BC
If the question means how much work is done to extend from its natural length to its final position it is simply 2(b(2a)^2)/2a...=4ab