The diagram shows a solid cone of base radius 4.2cm and slant height 13cm. (a) Show that the height of the cone is 10.2cm, correct to 3 significant figures. (b) Calculate the volume of the cone.

The radius ($r$), height ($h$), and slant height ($l$) of a cone form a right-angled triangle. We can use the Pythagorean theorem:

$$r^2 + h^2 = l^2$$

Given $r = 4.2$ cm and $l = 11$ cm.

$$(4.2)^2 + h^2 = (11)^2$$ $$17.64 + h^2 = 121$$ $$h^2 = 121 - 17.64$$ $$h^2 = 103.36$$ $$h = \sqrt{103.36}$$ $$h \approx 10.1666 \text{ cm}$$

$$h \approx 10.2 \text{ cm}$$

Thus, the height of the cone is $\boxed{\text{10.2 cm}}$ (correct to 3 significant figures).

The volume ($V$) of a cone is given by:

$$V = \frac{1}{3}\pi r^2 h$$

Using $r = 4.2$ cm and the more precise value of $h = \sqrt{103.36}$ cm from part (a):

$$V = \frac{1}{3} \times \pi \times (4.2)^2 \times \sqrt{103.36}$$ $$V = \frac{1}{3} \times \pi \times 17.64 \times \sqrt{103.36}$$ $$V \approx \frac{1}{3} \times 3.14159265 \times 17.64 \times 10.166612$$ $$V \approx 186.662 \text{ cm}^3$$

$$V \approx 187 \text{ cm}^3$$

The volume of the cone is $\boxed{\text{187 cm}^3}$.