Questions: A model rocket is launched with an initial upward velocity of 40 m/s. The rocket's height h (in meters) after t seconds is given by the following: h = 40t - 5t^2 Find all values of t for which the rocket's height is 15 meters. Round your answer(s) to the nearest hundredth. (If there is more than one answer, use the "or" button.) t = [] seconds

Solution Steps

We are given the height equation for the rocket: \[ h = 40t - 5t^2 \] We need to find the values of \( t \) when the height \( h \) is 15 meters. So, we set up the equation: \[ 15 = 40t - 5t^2 \]

Rearrange the equation to standard quadratic form: \[ -5t^2 + 40t - 15 = 0 \] Multiply through by -1 to simplify: \[ 5t^2 - 40t + 15 = 0 \]

We use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 5 \), \( b = -40 \), and \( c = 15 \).

Calculate the discriminant: \[ b^2 - 4ac = (-40)^2 - 4(5)(15) = 1600 - 300 = 1300 \]

Now, solve for \( t \): \[ t = \frac{-(-40) \pm \sqrt{1300}}{2(5)} = \frac{40 \pm \sqrt{1300}}{10} \]

Calculate the square root of 1300: \[ \sqrt{1300} \approx 36.0555 \]

So, the solutions are: \[ t = \frac{40 + 36.0555}{10} \approx 7.61 \] \[ t = \frac{40 - 36.0555}{10} \approx 0.39 \]

Final Answer