Cube surface area: \(6 \cdot 6^2 = 6 \cdot 36 = 216\).
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- An angel investor evaluates a biotech firm whose projected revenue follows the quadratic model \( R(x) = -2x^2 + 40x + 100 \), where \( x \) is years after launch. In which year does revenue peak?
- Vertex of parabola: \( x = -\frac{b}{2a} = -\frac{40}{2 \cdot (-2)} = \frac{40}{4} = 10 \).
- A game designer uses trigonometric scaling to morph a cube into a sphere. If the cube’s edge length is 6 units, by what factor must the surface area be scaled to match the sphere’s surface area?
- Cube volume: \(6^3 = 216\), so sphere radius \( r \) from volume: \( \frac{4}{3}\pi r^3 = 216 \) → \( r^3 = \frac{162}{\pi} \approx 51.57 \) → \( r \approx 3.72 \).
- Sphere surface area: \(4\pi r^2 \approx 4\pi (13.83) \approx 173.6\).
- Scale factor: \( \frac{173.6}{216} \approx 0.803 \).
