The thermal efficiency of an ideal air-standard Otto cycle is 0.5. The value of specific heat ratio of air is 1.4. The compression ratio of the cycle is ______________ (rounded off to 1 decimal place).

The correct answer is 5.7 (or 5.66).

Step-by-Step Explanation:

The thermal efficiency ($\eta$) of an ideal air-standard Otto cycle is given by the formula:
$$\eta =1-\frac{1}{r^{\gamma -1}}$$
where:
- $\eta$ is the thermal efficiency of the Otto cycle (given as 0.5)
- $r$ is the compression ratio
- $\gamma$ is the specific heat ratio of air (given as 1.4)

Substituting the given values into the formula:
$$0.5=1-\frac{1}{r^{1.4-1}}$$

Simplify the exponent in the denominator:
$$0.5=1-\frac{1}{r^{0.4}}$$

Rearranging the equation to solve for the term containing $r$:
$$\frac{1}{r^{0.4}}=1-0.5$$
$$\frac{1}{r^{0.4}}=0.5$$

Taking the reciprocal of both sides:
$$r^{0.4}=\frac{1}{0.5}=2$$

To find $r$, raise both sides to the power of $\frac{1}{0.4}$ (which is equivalent to 2.5):
$$r=2^{2.5}$$

Calculating the numerical value:
$$r=2^2\times 2^{0.5}=4\times \sqrt{2}\approx 4\times 1.4142=5.6568$$

Rounding the compression ratio:
- Rounded off to 1 decimal place: $r\approx 5.7$
- Rounded off to 2 decimal places: $r\approx 5.66$