Answer
String length $8ft$, tacks distance $6.9ft$
Work Step by Step
Step 1. Identify the given quantities: length of the plywood 8ft, width of the plywood 4ft
Step 2. The largest possible ellipse out of this plywood will have the standard equation $\frac{x^2}{a^2} +\frac{y^2}{b^2}=1 $ with $2a=8$ and $2b=4$, thus $a=4ft,b=2ft$
Step 3. Use the relationship $c^2=a^2-b^2$, we have $c=\sqrt {4^2-2^2}=2\sqrt 3$, thus the distance between the tacks is $2c=4\sqrt 3\approx6.93ft$
Step 4. At the vertex, the total length of the string can be found as $L=(a+c)+(a-c)=2a=8ft$
Step 5. Conclusions: the total length of the string is $8ft$ and the distance between the tacks is about $6.9ft$
