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A rectangular prism has a surface area of 94 square units and dimensions in the ratio 1:2:3. Find the volume.
Let dimensions be \(x\), \(2x\), \(3x\).
Surface area: \(2(x \cdot 2x + 2x \cdot 3x + 3x \cdot x) = 94\).
\(2(2x^2 + 6x^2 + 3x^2) = 94 \rightarrow 22x^2 = 94 \rightarrow x^2 = \frac{94}{22} = \frac{47}{11}\).
Volume: \(x \cdot 2x \cdot 3x = 6x^3 = 6 \left(\sqrt{\frac{47}{11}}\right)^3\).
Calculate: \(x^3 = \left(\frac{47}{11}\right)^{3/2}\), \(6 \times \left(\frac{47}{11}\right)^{3/2}\).
Approximate \(x \approx 2.02\), \(x^3 \approx 8.20\), volume \(\approx 6 \times 8.20 = 49.2\).
#### 6(47/11)^(3/2)
A ladder leans against a wall, reaching a height of 12 meters. If the base is 5 meters from the wall, what is the length of the ladder?
Use Pythagoras: \(c^2 = 12^2 + 5^2 = 144 + 25 = 169\).