Solve real-world problems involving projectile motion, area, revenue, and other quadratic models.
Quadratics model many real-world situations: 1. Projectile Motion (the most common!): [formula] where h = height (feet), t = time (seconds), v_0 = initial velocity, h_0 = initial height. (In metric: -4.9t^2 instead of -16t^2.) 2. Area Problems: length times width, often with a constraint. 3. Revenue/Profit: R = (text{price})(text{quantity}), where one depends on the other. 4. Number Problems:…
For h(t) = -16t^2 + v_0 t + h_0: | Question | How to find it | |---|---| | Initial height | h(0) = h_0 | | When does it hit the ground? | Solve h(t) = 0 | | Maximum height | t_{text{max}} = frac{v_0}{32}, then compute h(t_{text{max}}) | | When does it reach height k? | Solve h(t) = k |
Example: A ball is thrown upward from a 48-foot building with an initial velocity of 32 ft/s. h(t) = -16t^2 + 32t + 48. When does it hit the ground?
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