Minecraft is a video game where players build their own virtual environments. The maximum height of a building in Minecraft is 256 feet. The function
models the height an object, in feet, \textit{t} seconds after it is dropped from the tallest building in the game. Determine, algebraically, the number of seconds it takes for the object to hit the ground.
The Correct Answer and Explanation is:
Correct Answer: 4 seconds
Explanation:
The problem asks for the time it takes for an object to hit the ground. The height of the object is modeled by the function h(t) = -16t² + 256, where h(t) is the height in feet and t is the time in seconds.
When the object hits the ground, its height is 0 feet. To find the time this occurs, we must set the height function, h(t), equal to 0 and solve for the variable t.
The equation to solve is:
0 = -16t² + 256
To solve this equation algebraically, we can first isolate the t² term. We can add 16t² to both sides of the equation:
16t² = 256
Next, we divide both sides by 16 to solve for t²:
t² = 256 / 16
t² = 16
Finally, to find the value of t, we take the square root of both sides of the equation. This gives us two possible solutions:
t = ±√16
t = 4 or t = -4
In the context of this physics problem, time cannot be negative. The time measurement begins at t = 0 when the object is dropped. Therefore, the negative solution, t = -4, is not a valid answer for this scenario. We must choose the positive solution.
The only physically meaningful solution is t = 4. This means it takes 4 seconds for the object to hit the ground after being dropped from the 256-foot building.
