Questions: A diver jumps off a cliff 30 feet in the air into the ocean below. Her height above the water can be modeled by the function f(t)=-16 t^2+12 t+30, where t represents the time in seconds since she jumped. Use the given information to complete questions 6 through 10. 6. Write an equation that can be used to find the time it takes the diver to reach a height of 25 feet. 7. Solve the equation from question 6. Round to the nearest thousandth. 8. Are both solutions to this equation viable? Explain your thinking. 9. Interpret the solution to the equation in the context of the situation. 10. How long will it take before the diver hits the surface of the water? Round to the nearest thousandth.

Solution Steps

6. To find the time it takes for the diver to reach a height of 25 feet, set the height function equal to 25 and solve for \( t \).

7. Solve the quadratic equation obtained in question 6 using the quadratic formula. Round the solutions to the nearest thousandth.

8. Evaluate the solutions from question 7 to determine if both are viable in the context of the problem, considering the physical scenario.

To find the time \( t \) when the diver reaches a height of 25 feet, we set the height function equal to 25: \[ -16t^2 + 12t + 30 = 25 \] This simplifies to: \[ -16t^2 + 12t + 5 = 0 \]

Using the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = -16 \), \( b = 12 \), and \( c = 5 \), we find the solutions: \[ t = \frac{-12 \pm \sqrt{12^2 - 4 \cdot (-16) \cdot 5}}{2 \cdot (-16)} \] This results in: \[ t = \frac{3}{8} - \frac{\sqrt{29}}{8} \quad \text{and} \quad t = \frac{3}{8} + \frac{\sqrt{29}}{8} \]

Calculating the numerical values gives: \[ t_1 \approx -0.298 \quad \text{and} \quad t_2 \approx 1.048 \] Rounding to the nearest thousandth, we have: \[ t_1 \approx -0.298 \quad \text{and} \quad t_2 \approx 1.048 \]

Since time cannot be negative, the only viable solution is: \[ t \approx 1.048 \]

Final Answer