The equation $x\sin x + \cos x = 1.015$ has a positive root $\alpha$ close to zero. Use small-angle approximations for $\sin x$ and $\cos x$ to obtain an approximation to $\alpha$. Give your answer correct to 3 significant figures.

1. Substitute the small-angle approximations for $\sin x$ and $\cos x$ into the given equation.

Using $\sin x \approx x$ and $\cos x \approx 1 - \frac{1}{2}x^2$ for small $x$ (in radians):
$x(x) + \left(1 - \frac{1}{2}x^2\right) \approx 1.015$

2. Simplify the resulting algebraic expression to isolate $x^2$.

$x^2 + 1 - 0.5x^2 = 1.015$
$0.5x^2 + 1 = 1.015$
$0.5x^2 = 0.015$

3. Solve for the positive root and round to 3 significant figures.

$x^2 = \frac{0.015}{0.5} = 0.03$
Since $\alpha$ is a positive root, we take the positive square root:
$x = \sqrt{0.03} \approx 0.173205$
$\alpha = 0.173 \text{ (to 3 s.f.)}$