The correct answer is $\boxed{\text{B) 707 V}}$.
A step-up transformer increases the voltage in the secondary coil relative to the voltage in the primary coil. The turns ratio is the number of turns in the secondary coil divided by the number of turns in the primary coil. In this case, the turns ratio is 1:4, which means that the secondary coil has 4 times as many turns as the primary coil.
The peak voltage of a sine wave is equal to $\sqrt{2}$ times the RMS voltage. Therefore, the peak secondary voltage is equal to $\sqrt{2} \times 115 \text{ V} = 707 \text{ V}$.
Option A is incorrect because it is the RMS voltage of the secondary coil, not the peak voltage. Option C is incorrect because it is the RMS voltage of the primary coil, not the secondary coil. Option D is incorrect because it is the peak voltage of the primary coil, not the secondary coil. Option E is incorrect because it is not one of the possible answers.