Question

A ball is thrown from an initial height of 3 meters with an initial upward velocity of 30 m/s. The ball’s height h (in meters) after t seconds is given by the following. h=3+30t-5t^2 Find all values of t for which the ball’s height is 13 meters. Round your answer(s) to the nearest hundredth.

Answer 1

The values of t for which the ball's height is 13 meters is;

`\begin{gathered} t=0.35\text{ s} \\ or \\ t=5.65\text{ s} \end{gathered}`

Step-by-step explanation:

The function of the ball's height h (in meters) is given as;

`h=3+30t-5t^2`

the value of time t for which the ball's height is 13 meters, can be derived by substituting h=13 into the function of h.

`\begin{gathered} h=3+30t-5t^2 \\ 13=3+30t-5t^2 \\ 3+30t-5t^2=13 \end{gathered}`

subtract 13 from both sides and solve the quadratic equation;

`\begin{gathered} 3+30t-5t^2-13=13-13 \\ -5t^2+30t-10=0 \end{gathered}`

solving the quadratic equation, using the quadratic formula;

`\begin{gathered} t=(-b\pm√(b^2-4ac))/(2a) \\ t=(-30\pm√(30^2-4*-5*-10))/(2*-5) \\ t=(-30\pm√(900-200))/(-10) \\ t=(-30\pm√(700))/(-10) \\ t=0.3542=0.35 \\ or \\ t=5.64575=5.65 \end{gathered}`

The values of t for which the ball's height is 13 meters is;

`\begin{gathered} t=0.35\text{ s} \\ or \\ t=5.65\text{ s} \end{gathered}`