The problem asks to find the lateral area ($L$) and surface area ($S$) of a cone. The radius ($r$) of the base is $14$ mm and the slant height ($l$) is $24$ mm.

1. Problem Description

The problem asks to find the lateral area ($L$) and surface area ($S$) of a cone. The radius ($r$) of the base is $14$ mm and the slant height ($l$) is $24$ mm.

2. Solution Steps

First, we calculate the lateral area of the cone using the formula:

$L = \pi r l$

$L = \pi \times 14 \times 24$

$L = 336\pi$

$L \approx 1055.575$

Rounding to the nearest tenth, $L \approx 1055.6$ mm$^2$.

Next, we calculate the area of the base using the formula:

$B = \pi r^2$

$B = \pi (14)^2$

$B = 196\pi$

$B \approx 615.752$

Then, we calculate the surface area of the cone using the formula:

$S = L + B$

$S = \pi r l + \pi r^2$

$S = 336\pi + 196\pi$

$S = 532\pi$

$S \approx 1671.298$

Rounding to the nearest tenth, $S \approx 1671.3$ mm$^2$.

3. Final Answer

$L \approx 1055.6$ mm$^2$

$S \approx 1671.3$ mm$^2$