230. a ball is thrown into the air and the vertical position is given by (x(t)=-4.9t^{2}+25t + 5). use the…

230. a ball is thrown into the air and the vertical position is given by (x(t)=-4.9t^{2}+25t + 5). use the intermediate value theorem to show that the ball must land on the ground sometime between 5 sec and 6 sec after the throw.

230. a ball is thrown into the air and the vertical position is given by (x(t)=-4.9t^{2}+25t + 5). use the intermediate value theorem to show that the ball must land on the ground sometime between 5 sec and 6 sec after the throw.

Answer

Answer:

The ball must land on the ground between 5 sec and 6 sec.

Explanation:

Step1: Define the function

Let $x(t)=-4.9t^{2}+25t + 5$.

Step2: Evaluate at $t = 5$

$x(5)=-4.9\times5^{2}+25\times5 + 5=-4.9\times25+125 + 5=-122.5+125 + 5=7.5$.

Step3: Evaluate at $t = 6$

$x(6)=-4.9\times6^{2}+25\times6 + 5=-4.9\times36+150 + 5=-176.4+150 + 5=-21.4$.

Step4: Apply Intermediate - Value Theorem

The function $x(t)$ is continuous (it's a polynomial). Since $x(5)=7.5>0$ and $x(6)=-21.4<0$, by the Intermediate - Value Theorem, there exists a $c\in(5,6)$ such that $x(c) = 0$. When $x(t)=0$, the ball is on the ground. So the ball must land on the ground between 5 sec and 6 sec.