What is the maximum value of the AC voltage whose root-mean-square value is (a) 117 V.  V (b) 213 V. V

1. Break Down the Problem

To find the maximum value (peak voltage) of an AC voltage from its root-mean-square (RMS) voltage, we need to use the relationship between RMS voltage and peak voltage.

2. Relevant Concepts

The relationship between the RMS voltage ($V_{rms}$) and the maximum voltage ($V_{max}$) for an AC circuit is given by the formula:

$V_{max} = V_{rms} \times \sqrt{2}$

3. Analysis and Detail

Now we will calculate the maximum value for both cases.

Case (a):
Given $V_{rms} = 117 V$

$V_{max} = 117 V \times \sqrt{2} \approx 117 V \times 1.414 \approx 165.0 V$

Case (b):
Given $V_{rms} = 213 V$

$V_{max} = 213 V \times \sqrt{2} \approx 213 V \times 1.414 \approx 300.0 V$

4. Verify and Summarize

The calculations for both cases can be double-checked by relying on the precision of $\sqrt{2}$. The results for the peak voltage from the given RMS values are accurate.

Final Answer

(a) The maximum value of the AC voltage whose RMS value is $117 V$ is approximately 165.0 V.
(b) The maximum value of the AC voltage whose RMS value is $213 V$ is approximately 300.0 V.