Answer
The slanted cable has a tension of 1410 N.
The horizontal cable has a tension of 6210 N.
Work Step by Step
We can find the speed $v$ when the rotation rate is 28.0 rpm.
$v = (28.0~rpm)(\frac{1~min}{60~s})(\frac{(2\pi)(7.50~m)}{1~rev})$
$v = 22.0~m/s$
Let $T_1$ be the tension in the slanted cable. Note that the total weight of the passenger and the seat is 1080 N and the total mass $m$ is $\frac{1080~N}{g}$
$T_1~cos(40.0^{\circ}) = mg$
$T_1 = \frac{mg}{cos(40.0^{\circ})} = \frac{1080~N}{cos(40.0^{\circ})}$
$T_1 = 1410~N$
Let $T_2$ be the tension in the horizontal cable.
$T_2 + T_1~sin(40.0^{\circ}) = \frac{mv^2}{r}$
$T_2 = \frac{mv^2}{r}- T_1~sin(40.0^{\circ})$
$T_2 = \frac{(1080~N)(22.0~m/s)^2}{(7.50~m)(9.80~m/s^2)}- (1410~N)~sin(40.0^{\circ})$
$T_2 = 6210~N$
