Ouadratic Equations and Functions
Solving a word problem using a quadratic equation with irrational roots
A ball is thrown from a height of 64 meters with an initial downward velocity of $3 \mathrm{~m} / \mathrm{s}$. The ball's height $h$ (in meters) after $t$ seconds is given by the following
\[
h=64-3 t-5 t^{2}
\]
How long after the ball is thrown does it hit the ground?
Round your answer(s) to the nearest hundredth.
(If there is more than one answer, use the "or" button.)
\[
t=\square \text { seconds }
\]
$\square \propto \square$
Answer
Final Answer: \(\boxed{3.29}\) seconds.
Step 1 ：The problem gives us a quadratic equation in the form of \(ax^2 + bx + c = 0\). We can solve this equation using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). In this case, a = -5, b = -3, and c = 64. We will substitute these values into the quadratic formula to find the value of 't'.
Step 2 ：Substituting the values into the quadratic formula, we get two solutions for 't': 3.29 and -3.89. However, time cannot be negative, so we discard the negative solution.
Step 3 ：The ball hits the ground 3.29 seconds after it is thrown.
Step 4 ：Final Answer: \(\boxed{3.29}\) seconds.
