Question

eoc style question on a vacation in puerto rico, rose jumped off a cliff into a river in the el yunque forest reserve. her height as a function of time could be modeled by the function ( h(t) = -16t^2 + 16t + 480 ) where ( t ) is the time after jumping in seconds and ( h ) is the height in feet. how many seconds did it take rose to hit the water? round your answer to the nearest hundredth. (\boxed{quad}) seconds

Explanation:

Step1: Set \( h(t) = 0 \)

When Rose hits the water, her height \( h(t) = 0 \). So we set up the equation:
\( -16t^2 + 16t + 480 = 0 \)

Step2: Simplify the equation

Divide the entire equation by -16 to simplify:
\( t^2 - t - 30 = 0 \)

Step3: Solve the quadratic equation

We use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) for a quadratic equation \( ax^2 + bx + c = 0 \). Here, \( a = 1 \), \( b = -1 \), and \( c = -30 \).
First, calculate the discriminant \( \Delta = b^2 - 4ac = (-1)^2 - 4(1)(-30) = 1 + 120 = 121 \)
Then, find the roots:
\( t = \frac{-(-1) \pm \sqrt{121}}{2(1)} = \frac{1 \pm 11}{2} \)
We have two solutions:
\( t_1 = \frac{1 + 11}{2} = \frac{12}{2} = 6 \)
\( t_2 = \frac{1 - 11}{2} = \frac{-10}{2} = -5 \)
Since time cannot be negative, we discard \( t_2 \).

Answer:

6.00