Question

a ball is thrown from an initial height of 2 feet with an initial upward velocity of 23 ft/s. the balls height h (in feet) after t seconds is given by the following.
h = 2 + 23t - 16t²
find all values of t for which the balls height is 10 feet.
round your answer(s) to the nearest hundredth.
(if there is more than one answer, use the or button.)

Explanation:

Step1: Set up the equation

Set \(h = 10\) in the equation \(h=2 + 23t-16t^{2}\). So, \(10=2 + 23t-16t^{2}\).
Rearrange it to the standard quadratic form \(ax^{2}+bx + c = 0\). We get \(16t^{2}-23t + 8=0\). Here, \(a = 16\), \(b=-23\), \(c = 8\).

Step2: Use the quadratic formula

The quadratic formula is \(t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).
Substitute \(a = 16\), \(b=-23\), \(c = 8\) into the formula.
First, calculate the discriminant \(\Delta=b^{2}-4ac=(-23)^{2}-4\times16\times8=529 - 512=17\).
Then, \(t=\frac{23\pm\sqrt{17}}{2\times16}\).
Since \(\sqrt{17}\approx4.123\), \(t_{1}=\frac{23 + 4.123}{32}\approx\frac{27.123}{32}\approx0.85\) and \(t_{2}=\frac{23-4.123}{32}\approx\frac{18.877}{32}\approx0.59\).

Answer:

\(t = 0.59\) or \(t = 0.85\) seconds