The surface area of a cube, \( A \), is represented by \( A=6 s^{2} \), where \( s \) is the length of each edge of the cube. If the surface area of a cube is 288 , which of the following equations gives each edge's length and provides correct reasoning? The equation \( s=2 \sqrt{24} \) gives each edge's length because \( \sqrt{48}=\sqrt{2 \cdot 24} \). The equation \( s=4 \sqrt{3} \) gives each edge's length because \( \sqrt{48}=\sqrt{16 \cdot 3} \). The equation \( s=5 \sqrt{12} \) gives each edge's length because \( \sqrt{48}=\sqrt{36+12} \). gives each edge's length because \( \sqrt{48}=\sqrt{25+23} \).

Let's unpack the solution here! If the surface area of the cube is \( A = 288 \), then we can set up the equation: \[ 6s^2 = 288. \] Dividing both sides by 6, we find \( s^2 = 48 \). Now, taking the square root gives us \( s = \sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3} \). So, the equation \( s = 4\sqrt{3} \) is indeed the right fit, as it properly breaks down the square root! Now, let’s not overlook those pesky common mistakes! A frequent slip is in simplifying square roots. When faced with \( \sqrt{48} \), many leap straight to \( 7 \) or mishandle the factors, ignoring proper factorization. Always remember, simplifying inside the square root leads to the clearest path for finding edge lengths or other dimensions!