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Home » Guardians of the Globe Exposed: The Secret Heroes Saving the Planet! - Leaselab

Guardians of the Globe Exposed: The Secret Heroes Saving the Planet! - Leaselab

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 Calculating \( 6! \):

📌 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720

📌 Therefore, the number of ways to arrange the participants around the table is \(\boxed{720}\).Question: An equilateral triangle-shaped solar panel array has an area of $ 36\sqrt{3} \, \text{cm}^2 $. If each side is reduced by 4 cm, by how many square centimeters does the area decrease?

📌 Solution: The area of an equilateral triangle is given by $ \frac{\sqrt{3}}{4} s^2 $, where $ s $ is the side length. Setting $ \frac{\sqrt{3}}{4} s^2 = 36\sqrt{3} $, we solve for $ s^2 = 144 $, so $ s = 12 \, \text{cm} $. The new side length is $ 12 - 4 = 8 \, \text{cm} $. The new area is $ \frac{\sqrt{3}}{4} \times 8^2 = 16\sqrt{3} \, \text{cm}^2 $. The decrease in area is $ 36\sqrt{3} - 16\sqrt{3} = 20\sqrt{3} \, \text{cm}^2 $. \boxed{20\sqrt{3}}

📌 Question: A spherical data storage unit has a radius of $ y $ units, and a hemispherical backup dome has a radius of $ 3y $ units. What is the ratio of the storage unit's volume to the backup dome's volume?

📌 Solution: The volume of a sphere is $ \frac{4}{3}\pi y^3 $, and the volume of a hemisphere is $ \frac{2}{3}\pi (3y)^3 = \frac{2}{3}\pi \times 27y^3 = 18\pi y^3 $. The ratio is $ \frac{\frac{4}{3}\pi y^3}{18\pi y^3} = \frac{4}{54} = \frac{2}{27} $. \boxed{\dfrac{2}{27}}

📌 Question: A triangular geological fault line has side lengths of 10 cm, 13 cm, and 15 cm. Compute the length of the shortest altitude.

📌 Solution: First, compute the area using Heron's formula. The semi-perimeter $ s = \frac{10 + 13 + 15}{2} = 19 \, \text{cm} $. The area is $ \sqrt{19(19-10)(19-13

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