Questions: A small rock sits on the edge of a tall building. A strong wind blows the rock off the edge. The distance, in feet, between the rock and the ground t seconds after the rock leaves the edge is given by d=-16t^2-5t+470. If the answer is not an integer, enter it as a decimal. Round to the nearest hundredth, if needed. How many seconds after the rock leaves the edge is it 449 feet from the ground? seconds How many seconds after the rock leaves the edge does it hit the ground? seconds

Solution Steps

We need to find the time \( t \) when the rock is 449 feet from the ground. The distance function is given by: \[ d = -16t^2 - 5t + 470 \] We set \( d = 449 \) and solve for \( t \): \[ 449 = -16t^2 - 5t + 470 \]

Rearrange the equation to standard quadratic form: \[ -16t^2 - 5t + 470 - 449 = 0 \] \[ -16t^2 - 5t + 21 = 0 \]

Use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = -16 \), \( b = -5 \), and \( c = 21 \): \[ t = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(-16)(21)}}{2(-16)} \] \[ t = \frac{5 \pm \sqrt{25 + 1344}}{-32} \] \[ t = \frac{5 \pm \sqrt{1369}}{-32} \] \[ t = \frac{5 \pm 37}{-32} \]

Calculate the two possible values for \( t \): \[ t_1 = \frac{5 + 37}{-32} = \frac{42}{-32} = -1.3125 \] \[ t_2 = \frac{5 - 37}{-32} = \frac{-32}{-32} = 1 \]

Since time cannot be negative, we discard \( t_1 \): \[ t = 1 \]

We need to find the time \( t \) when the rock hits the ground, i.e., when \( d = 0 \): \[ 0 = -16t^2 - 5t + 470 \]

Use the quadratic formula again with \( a = -16 \), \( b = -5 \), and \( c = 470 \): \[ t = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(-16)(470)}}{2(-16)} \] \[ t = \frac{5 \pm \sqrt{25 + 30080}}{-32} \] \[ t = \frac{5 \pm \sqrt{30105}}{-32} \] \[ t = \frac{5 \pm 173.5290}{-32} \]

Calculate the two possible values for \( t \): \[ t_1 = \frac{5 + 173.5290}{-32} = \frac{178.5290}{-32} = -5.5790 \] \[ t_2 = \frac{5 - 173.5290}{-32} = \frac{-168.5290}{-32} = 5.2665 \]

Since time cannot be negative, we discard \( t_1 \): \[ t = 5.2665 \]

Final Answer