Let’s go through the problems step-by-step. Each question will be solved in detail.

Question 1: Soccer Ball Problem

The equation for the height is $h(t) = -16t^2 + 90t$.
(a) How many seconds will it take for the ball to reach its maximum height?

To find the time $t$ to reach the maximum height, use the formula for the vertex of a parabola: $t = -\frac{b}{2a}$ where $a = -16$ and $b = 90$.

$t = -\frac{90}{2(-16)} = \frac{90}{32} = 2.8125 \, \text{seconds.}$

(b) What is the maximum height of the ball?

Substitute $t = 2.8125$ back into $h(t)$: $h(2.8125) = -16(2.8125)^2 + 90(2.8125)$ $h(2.8125) = -16(7.91015625) + 90(2.8125) = -126.5625 + 253.125 = 126.5625 \, \text{feet.}$

(c) What is the domain and range of the function?

• Domain: The ball starts at $t = 0$ and hits the ground when $h(t) = 0$. Solve: $-16t^2 + 90t = 0$ Factor: $t(-16t + 90) = 0 \implies t = 0 \, \text{or} \, t = \frac{90}{16} = 5.625 \, \text{seconds.}$ Domain: $[0, 5.625]$.

• Range: The minimum height is 0 (at $t = 0$ and $t = 5.625$). The maximum height is 126.5625 feet.
  Range: $[0, 126.5625]$.

Question 2: Apple Launched Upward

The equation for the height is $h(t) = -16t^2 + 64t + 80$.
Find the height of the apple after 3 seconds.

Substitute $t = 3$ into $h(t)$: $h(3) = -16(3)^2 + 64(3) + 80$ $h(3) = -16(9) + 192 + 80 = -144 + 192 + 80 = 128 \, \text{feet.}$

Question 3: Toy Rocket

The equation is $h(t) = -16t^2 + 128t + 12$.
How long will it take for the rocket to reach the ground?

Solve for $h(t) = 0$: $-16t^2 + 128t + 12 = 0$ Divide through by -4: $4t^2 - 32t - 3 = 0$ Use the quadratic formula: $t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a = 4$, $b = -32$, and $c = -3$: $t = \frac{-(-32) \pm \sqrt{(-32)^2 - 4(4)(-3)}}{2(4)} = \frac{32 \pm \sqrt{1024 + 48}}{8}$ $t = \frac{32 \pm \sqrt{1072}}{8} = \frac{32 \pm 32.74}{8}$ $t = \frac{64.74}{8} = 8.0925 \, \text{seconds (approx.) or } \, t = \frac{-0.74}{8} \, \text{(discarded as negative)}.$

Question 4: Penny Dropped

The equation is $h(t) = -16t^2 + 1250$.
How long would it take the penny to strike a 6-foot tall person?

Solve for $h(t) = 6$: $-16t^2 + 1250 = 6$ $-16t^2 = 6 - 1250 = -1244$ $t^2 = \frac{1244}{16} = 77.75$ $t = \sqrt{77.75} \approx 8.82 \, \text{seconds.}$

Let me know if you'd like further details on any of the solutions!

Follow-up questions:

1. How do we calculate the total time in projectile motion problems?
2. Why do we use the vertex formula for maximum height?
3. How do we interpret the domain and range of these quadratic functions?
4. How does gravity ($-16t^2$) affect the shape of the projectile motion?
5. What would happen if the initial velocity was negative?

Tip: Always check if the equation is factored or needs the quadratic formula to save time during tests!