- Define the quadratic equation for height: \(h = -16t^2 + 255\).
- Set \(h = 0\) and identify coefficients for the quadratic formula: \(a = -16, b = 0, c = 255\).
- Solve for \(t\) using the quadratic formula: \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \approx 3.99 \text{ s}\).
- Match the result to the provided options, identifying the final answer as \(\boxed{c}\).
Explanation
- Problem Analysis and Setup
The height of the object at any time \(t\) is given by the equation \(h = -16t^2 + h_0\). We are given that the initial height \(h_0 = 255 \text{ ft}\), so the equation becomes \(h = -16t^2 + 255\). To find the time it takes the rock to reach the canyon floor, we set the height \(h\) to \(0\), resulting in the equation \(-16t^2 + 255 = 0\). This is a quadratic equation of the form \(at^2 + bt + c = 0\), where \(a = -16\), \(b = 0\), and \(c = 255\).
- Applying the Quadratic Formula
We solve the quadratic equation \(-16t^2 + 255 = 0\) using the quadratic formula, which is \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substituting our coefficients \(a = -16\), \(b = 0\), and \(c = 255\), we get:
\[t = \frac{-0 \pm \sqrt{0^2 - 4(-16)(255)}}{2(-16)}\]
\[t = \frac{\pm \sqrt{64 \times 255}}{-32}\]
\[t = \frac{\pm \sqrt{16320}}{-32}\]
Calculating the square root, we have \(\sqrt{16320} \approx 127.75\).
Thus, \(t \approx \frac{\pm 127.75}{-32}\).
Since time must be positive, we choose the solution that yields a positive value for \(t\):
\[t \approx \frac{-127.75}{-32} \approx 3.99 \text{ s}.\]
Rounding to the nearest whole number, we find \(t \approx 4 \text{ s}\).
- Conclusion and Selection
Comparing our result \(t \approx 3.99 \text{ s}\) with the given options:
A\(t = 16 \text{ s}\)
B\(t \approx 0.87 \text{ s}\)
C\(t \approx 4 \text{ s}\)
D\(t = 8.5 \text{ s}\)
The calculated time \(t \approx 4 \text{ s}\) matches option c. Therefore, it takes approximately \(4\) seconds for the rock to hit the canyon floor. The final answer is \(\boxed{c}\).
Examples
This quadratic model is used extensively in physics to calculate the time of fall for objects under gravity. For example, if you drop a ball from a balcony or estimate the time an object takes to land, you are essentially solving a quadratic equation. Understanding how to apply the quadratic formula allows you to predict motion and time events accurately in real-world scenarios.