The given problem asks us to determine the maximum height a person can jump based on the function $h(t) = -16t^2 + 12t$, where $h(t)$ represents the height (in feet) above the ground after $t$ seconds. It also asks whether the person can dunk the basketball by reaching a height of 2.5 ft.

Step 1: Find the maximum height

The function $h(t) = -16t^2 + 12t$ is a quadratic equation. The maximum height occurs at the vertex of the parabola. For a quadratic function $ax^2 + bx + c$, the time at which the vertex occurs is given by: $t = -\frac{b}{2a}$ Here, $a = -16$ and $b = 12$. Substituting into the formula: $t = -\frac{12}{2(-16)} = \frac{12}{32} = 0.375 \, \text{seconds}$

Step 2: Calculate $h(t)$ at $t = 0.375$

Substitute $t = 0.375$ into $h(t)$: $h(0.375) = -16(0.375)^2 + 12(0.375)$ $h(0.375) = -16(0.140625) + 12(0.375)$ $h(0.375) = -2.25 + 4.5 = 2.25 \, \text{feet}$

The maximum height the person's feet can reach is 2.25 feet.

Step 3: Compare with the required height

To dunk the basketball, the person needs to jump 2.5 feet. Since the maximum height (2.25 feet) is less than 2.5 feet, the person will not be able to dunk the basketball.

Final Answer:

• Maximum height: 2.25 feet.
• Will the person dunk? No, they cannot dunk the basketball.

Would you like me to provide detailed explanations for any step or additional help? Here are 5 related questions to expand on this topic:

1. How would the maximum height change if the initial velocity increased to 14 ft/s instead of 12 ft/s?
2. What is the significance of the negative sign in the quadratic term of the height function?
3. Can you explain how to graph this quadratic function and interpret its features?
4. How would you determine the total time the person is in the air using this height function?
5. What physical factors could influence the height function and make it more realistic?

Tip: To solve quadratic functions efficiently, always identify the vertex formula and check if the parabola opens upwards ($a > 0$) or downwards ($a < 0$).