Test Questions on Quadratic Optimization and Projectile Motion.

📚 Quick Study Guide

• 📈 Quadratic Optimization: This involves finding the maximum or minimum value of a quadratic function. The vertex of the parabola represents the optimal value. For a quadratic function in the form $f(x) = ax^2 + bx + c$, the x-coordinate of the vertex is given by $x = -\frac{b}{2a}$. If $a > 0$, the parabola opens upwards (minimum value); if $a < 0$, it opens downwards (maximum value).
• 🎯 Projectile Motion: This is the motion of an object thrown or projected into the air, subject only to gravity. Key formulas include:
• 🚀 Horizontal motion: $x = v_{0x}t$ (constant velocity)
• vertical motion: $y = v_{0y}t - \frac{1}{2}gt^2$ (constant acceleration due to gravity, $g \approx 9.8 m/s^2$)
• Initial velocity components: $v_{0x} = v_0 \cos(\theta)$ and $v_{0y} = v_0 \sin(\theta)$, where $v_0$ is the initial speed and $\theta$ is the launch angle.
• Time of flight: $T = \frac{2v_{0y}}{g}$
• Range: $R = \frac{v_0^2 \sin(2\theta)}{g}$
• 💡 Important Considerations: Air resistance is often ignored in introductory projectile motion problems. The maximum range is achieved when the launch angle is 45 degrees (assuming flat ground and no air resistance).
• 📐 Completing the Square: A technique to rewrite a quadratic equation in vertex form, $f(x) = a(x - h)^2 + k$, where (h, k) is the vertex. This is very useful for finding maximum/minimum values.

Practice Quiz

1. A farmer wants to fence off a rectangular area next to a river, using the river as one side of the rectangle. He has 100 meters of fence. What dimensions maximize the area enclosed?
   1. A. 25m x 50m
   2. B. 20m x 60m
   3. C. 30m x 40m
   4. D. 50m x 50m
2. A projectile is launched with an initial velocity of 20 m/s at an angle of 30 degrees above the horizontal. What is the maximum height reached by the projectile? (Assume $g = 9.8 m/s^2$)
   1. A. 2.55 m
   2. B. 5.10 m
   3. C. 10.2 m
   4. D. 20.4 m
3. Which launch angle (relative to the horizontal) will result in the maximum range for a projectile, assuming flat ground and negligible air resistance?
   1. A. 30 degrees
   2. B. 45 degrees
   3. C. 60 degrees
   4. D. 90 degrees
4. A quadratic function is given by $f(x) = -2x^2 + 8x - 5$. What is the maximum value of this function?
   1. A. -5
   2. B. 3
   3. C. 8
   4. D. 11
5. A ball is thrown vertically upwards with an initial velocity of 15 m/s. How long will it take for the ball to reach its maximum height? (Assume $g = 9.8 m/s^2$)
   1. A. 0.77 s
   2. B. 1.53 s
   3. C. 2.30 s
   4. D. 3.06 s
6. What is the range of a projectile launched at an angle of 45 degrees with an initial velocity of 30 m/s? (Assume $g = 9.8 m/s^2$)
   1. A. 45.9 m
   2. B. 91.8 m
   3. C. 137.7 m
   4. D. 183.6 m
7. The height $h(t)$ of a projectile is modeled by the equation $h(t) = -5t^2 + 30t + 2$. What is the maximum height the projectile reaches?
   1. A. 2 m
   2. B. 30 m
   3. C. 45 m
   4. D. 47 m