The maximum amplitude for an amplitude modulated wave is found to be 12 V while the minimum amplitude is found to be 3 V. The modulation index is 0.6$$x$$ where $$x$$ is _________.

For an amplitude modulated (AM) signal, the envelope shows a highest (maximum) value and a lowest (minimum) value of the instantaneous amplitude. The quantity that tells us how strongly the carrier is being modulated by the message is called the modulation index, usually denoted by $$m$$.

First we recall the standard formula that relates the modulation index $$m$$ to the maximum envelope voltage $$E_{\max}$$ and the minimum envelope voltage $$E_{\min}$$:

$$m \;=\; \dfrac{E_{\max}-E_{\min}}{E_{\max}+E_{\min}}.$$

This formula can be remembered as “difference over sum,” and it comes directly from writing the upper and lower envelopes of an AM wave as $$E_c(1+m)$$ and $$E_c(1-m)$$, where $$E_c$$ is the unmodulated carrier amplitude. Subtracting and adding those two expressions gives the above relation.

Now we substitute the numerical values given in the question. The maximum envelope voltage is $$E_{\max}=12\text{ V}$$ and the minimum envelope voltage is $$E_{\min}=3\text{ V}$$. Putting these into the formula gives

$$m \;=\; \dfrac{12\text{ V}-3\text{ V}}{12\text{ V}+3\text{ V}}.$$

Carrying out the subtraction in the numerator, we have

$$m \;=\; \dfrac{9\text{ V}}{12\text{ V}+3\text{ V}}.$$

Next we add the voltages in the denominator:

$$m \;=\; \dfrac{9\text{ V}}{15\text{ V}}.$$

Since both numerator and denominator carry the unit “volt,” the units cancel, leaving a pure number:

$$m \;=\; \dfrac{9}{15}.$$

To simplify the fraction, we divide both numerator and denominator by their highest common factor, which is 3:

$$m \;=\; \dfrac{9\div 3}{15\div 3} \;=\; \dfrac{3}{5}.$$

Writing the fraction $$\dfrac{3}{5}$$ in decimal form, we remember that $$\dfrac{3}{5}=0.6$$. Hence we have

$$m = 0.6.$$

The statement in the problem says that “the modulation index is $$0.6x$$.” We have just found that the modulation index equals $$0.6$$ itself. Therefore, comparing the two expressions,

$$0.6x = 0.6.$$

To get the value of $$x$$, we divide both sides of the equation by $$0.6$$:

$$x = \dfrac{0.6}{0.6} = 1.$$

So, the answer is $$1$$.