Question: User: A square-based pyramid is shown below. It has four identical triangular faces.
Jade has some paint that will cover an area of 5580 mm²
How many pyramids identical to the one shown below can Jade paint completely?
15 mm
16 mm
Graph/shape: a square-based pyramid with four triangular faces; the base edge is labeled 16 mm and the slanted face height is labeled 15 mm, with the measurement 15 mm near the center/front face and 16 mm along the bottom base edge; position_hint: center
1. **State the problem:**
We have a square-based pyramid with a base edge of $16$ mm and triangular faces with a slant height of $15$ mm. Jade has paint that covers $5580$ mm². We want to find how many such pyramids Jade can paint completely.
2. **Find the surface area of one pyramid:**
The pyramid has one square base and four identical triangular faces.
- Area of the square base:
$$\text{Area}_{base} = 16 \times 16 = 256 \text{ mm}^2$$
- Area of one triangular face:
The base of each triangle is $16$ mm and the height (slant height) is $15$ mm.
$$\text{Area}_{triangle} = \frac{1}{2} \times 16 \times 15 = 8 \times 15 = 120 \text{ mm}^2$$
- Total area of four triangular faces:
$$4 \times 120 = 480 \text{ mm}^2$$
- Total surface area of the pyramid:
$$256 + 480 = 736 \text{ mm}^2$$
3. **Calculate how many pyramids Jade can paint:**
Jade can paint an area of $5580$ mm².
Number of pyramids:
$$\frac{5580}{736}$$
4. **Simplify the fraction:**
$$\frac{5580}{736} = \frac{\cancel{4} \times 1395}{\cancel{4} \times 184} = \frac{1395}{184} \approx 7.58$$
5. **Interpretation:**
Jade can paint $7$ complete pyramids because $7.58$ means she can paint 7 full pyramids and part of the 8th.
**Final answer:**
Jade can paint **7** pyramids completely.