Question: A square-based pyramid is shown below. It has four identical triangular faces. Liam has some paint that will cover an area of 7460 mm². How many pyramids identical to the one shown below can Liam paint completely? Graph/shapes: a square-based pyramid diagram is shown near the bottom-center of the page, with pink triangular faces and a dashed interior edge; position_hint ∈ {center, bottom-center}. 14mm 24mm Not drawn accurately Shape: a pyramid in the center with a base edge labeled 14mm and a vertical height/perpendicular line labeled 24mm; the figure is not drawn accurately.
1. **State the problem:** We need to find how many identical square-based pyramids Liam can paint completely with 7460 mm² of paint.
2. **Identify the dimensions:** The pyramid has a square base with side length $14\text{ mm}$ and a vertical height (perpendicular height from base to apex) of $24\text{ mm}$.
3. **Calculate the area of one pyramid:** The total surface area of a square-based pyramid is the sum of the base area and the areas of the four triangular faces.
4. **Calculate the base area:**
$$\text{Base area} = 14 \times 14 = 196\text{ mm}^2$$
5. **Calculate the slant height of the triangular faces:**
The slant height $l$ is the height of each triangular face, which can be found using the Pythagorean theorem.
The half base length is:
$$\frac{14}{2} = 7\text{ mm}$$
The vertical height is $24\text{ mm}$.
So,
$$l = \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625} = 25\text{ mm}$$
6. **Calculate the area of one triangular face:**
$$\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 14 \times 25 = 7 \times 25 = 175\text{ mm}^2$$
7. **Calculate the total area of the four triangular faces:**
$$4 \times 175 = 700\text{ mm}^2$$
8. **Calculate the total surface area of the pyramid:**
$$196 + 700 = 896\text{ mm}^2$$
9. **Calculate how many pyramids Liam can paint:**
$$\frac{7460}{896} \approx 8.326\ldots$$
Since Liam can only paint whole pyramids completely, he can paint **8 pyramids**.
**Final answer:** Liam can paint 8 identical pyramids completely.