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: Recall Intermediate Value Theorem

If (y = f(x)) is continuous on ([a,b]), and (k) is a number between (f(a)) and (f(b)), then there exists at least one number (c\in(a,b)) such that (f(c)=k).

Step2: Determine the function and interval

The height - function is (x(t)=- 4.9t^{2}+25t + 5), and the interval is ([5,6]). The ball is on the ground when (x(t)=0).

Step3: Evaluate (x(t)) at (t = 5)

[ \begin{align*} x(5)&=-4.9\times5^{2}+25\times5 + 5\ &=-4.9\times25+125 + 5\ &=-122.5+125 + 5\ &=7.5 \end{align*} ]

Step4: Evaluate (x(t)) at (t = 6)

[ \begin{align*} x(6)&=-4.9\times6^{2}+25\times6 + 5\ &=-4.9\times36+150 + 5\ &=-176.4+150 + 5\ &=-21.4 \end{align*} ]

Step5: Apply Intermediate Value Theorem

Since (x(t)) is a polynomial, it is continuous on ([5,6]). We have (x(5)=7.5>0) and (x(6)=-21.4<0). The value (k = 0) is between (x(5)) and (x(6)). So, by the Intermediate - Value Theorem, there exists a (t\in(5,6)) such that (x(t)=0), which means the ball lands on the ground between 5 sec and 6 sec.