Question

A glass candleholder in the shape of a triangular pyramid has a volume of 2,010 mm³. The base of the holder has a height of 13.4 mm and a base of 30 mm. What is the area

of the candleholder?

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the area of the candleholder, we first need to calculate the height of the triangular pyramid.

The formula for the volume of a triangular pyramid is V = (1/3) * base area * height.
Thus, 2,010 mm³ = (1/3) * (1/2 * base * height) * height.

Solving for height gives us:
2,010 = (1/3) * (1/2 * 30 * 13.4) * height
6,030 = (1/3) * 201 * height
6,030 = 67 * height
height = 90 mm

Now that we have the height, we can calculate the area of the candleholder.
The formula for the area of a triangular pyramid is:
Area = base area + (1/2) * perimeter * slant height

The base area of the triangular pyramid is:
Area_base = (1/2) * base * height = (1/2) * 30 * 13.4 = 201 mm²

The slant height of the triangular pyramid can be calculated using the Pythagorean theorem:
Slant height = sqrt(base² + height²) = sqrt(30² + 90²) = sqrt(900 + 8100) = sqrt(9000) = 30√10 mm

Now, the area of the candleholder is:
Area = 201 + (1/2) * (30 + 30 + 30√10) * 90
Area = 201 + (1/2) * 90 * 90 + 45√10
Area = 201 + 4050 + 45√10
Area ≈ 4,251.82 mm²

Therefore, the area of the candleholder is approximately 4,251.82 mm².