30.86 ohms
Using ohms law I=V/R, and watts being W=VI, you can create the formula W=V^2/R by substituting V/R for I in the watts formula:
W = V^2/R
Plug in the known values of 90 watts and 72 ohms.
90 = V^2/72
Solve for V by first multiplying both sides by 72
6480 = V^2
Then take the square root of both sides
80.5 = V
Using ohms law, you can calculate the amperage passing through the coil
I = V/R = 80.5/72 = 1.118 amperes
Now you need to figure out the total resistance needed to allow for 1.118 amperes with a voltage of 115 volts. Ohms law again.
I = V/R
1.118 = 115/R
Multiply both sides by R
1.118 R = 115
Divide both sides by 1.118
R = 115 / 1.118 = 102.86
Now the resistance for the rheostat is 102.86 minus the resistance of the coil. Giving
R = 102.86 - 72 = 30.86 ohms.
The rheostat has to have a resistance of 30.86 ohms
