Niko has an outdoor play tent in the form of a regular triangular pyramid, covered in fabric on all four sides. The surface area of the tent is 100 ft.2

Question

, the base is 6 ft., and the slant height is 8 ft.

Answer 1

To analyze Niko's outdoor play tent, which is in the form of a regular triangular pyramid, we can use the given dimensions to confirm the calculations for the surface area.

Here are the key components to consider:

1. Base Area (A_base): The base of the pyramid is an equilateral triangle. The formula for the area of an equilateral triangle with side length $s$ is: $$A_{base}=\frac{\sqrt{3}}{4}{s}^2$$ Given $s=6$ ft: $$A_{base}=\frac{\sqrt{3}}{4}\times 6^2=\frac{\sqrt{3}}{4}\times 36=9\sqrt{3}\approx 15.59{\text{ ft}}^2$$

2. Lateral Surface Area (A_lateral): The lateral surface area of a triangular pyramid with base area $A_{base}$ and slant height $l$ (the height from the apex to the midpoint of a side of the base) can be calculated as follows: $$A_{lateral}=\frac{1}{2}\times \text{perimeter}\times \text{slant height}$$ The perimeter $P$ of the base (an equilateral triangle) is: $$P=3s=3\times 6=18\text{ ft}$$ Plugging in the values: $$A_{lateral}=\frac{1}{2}\times 18\times 8=72{\text{ ft}}^2$$

3. Total Surface Area (A_total): The total surface area of the triangular pyramid is the sum of the base area and the lateral surface area: $$A_{total}=A_{base}+A_{lateral}$$ $$A_{total}=9\sqrt{3}+72\approx 15.59+72\approx 87.59{\text{ ft}}^2$$

However, the information states that the surface area is $100$ ft². Let's do a final check of whether this area aligns with your information or setup.

Given the values, if we need to reach a total surface area of $100$ ft², we could conclude that either the slant height needs to be adjusted, or there might be additional factors affecting the overall measurements not taken into account.

Let me know if you need further calculations or clarification!