Answer
$x_1\approx1.222\ ft$ and $x_2\approx7.528\ ft$
Work Step by Step
Step 1. Use the figure given in the Exercise and use the Pythagorean Theorem, we have the left side cable length $L=\sqrt {x^2+6^2}$, the right side cable length $R=\sqrt {(10-x)^2+8^2}$
Step 2. The total length of the cable is $18$ feet, we have $L+R=18$ and $\sqrt {x^2+6^2}+\sqrt {(10-x)^2+8^2}=18$
Step 3. Rewrite the above equation as $\sqrt {(10-x)^2+8^2}=18-\sqrt {x^2+6^2}$ and take the square on both sides to get $(10-x)^2+8^2=18^2+x^2+6^2-36\sqrt {x^2+6^2}$ or $9\sqrt {x^2+6^2}=49+5x$
Step 4. Take the square again on the above equation to get $81(x^2+6^2)=25x^2+490x+49^2$ which is equivalent to $56x^2-490x+515=0$
Step 5. Solve the above equation graphically as shown in the figure to get $x_1\approx1.222\ ft$ and $x_2\approx7.528\ ft$
