The correct option is \(\boxed{\bf\ option D}\) i.e., \(\boxed{1\leq t\leq 3}\).
Further explanation:
Given:
The height of the ball is determined by the equation \(h(t)=-16t^{2}+64t^{2}+4\).
Calculation:
Consider \(t\) as the interval of time and \(h\) as the height of the time.
The equation is given as follows:
\(\boxed{h(t)=-16t^{2}+64t^{2}+4}\)
Further solve the above equation as follows:
\(\begin{aligned}h(t)&=-16t^{2}+64t^{2}+4\\&=48t^{2}+4\end{aligned}\)
We have to check the interval in which height of the ball is greater than or equal to \(52\text{ feet}\).
Check the option A by substituting \(t=1\) in the equation \(h(t)=-16t^{2}+64t^{2}+4\) as follows:
\(\begin{aligned}h(1)&=48\cdot (1)^{2}+4\\&=18+4\\&=52\end{aligned}\)
The height of the ball is \(52\text{ feet}\) at \(t=1\).
The option A is not full filling the condition completely as the height is not greater than \(52\text{ feet}\).
This implies that option A is incorrect.
Now, move to the option B that is \(t\leq 1\).
Check the second option by substituting \(t=0\) in the equation \(h(t)=48t^{2}+4\) as follows:
\(\begin{aligned}h(0)&=48\cdot (0)^{2}+4\\&=0+4\\&=4\end{aligned}\)
The height of the ball is \(4\text{ feet}\) at time \(t=0\).
The inequality for time \(t\leq 1\) is not satisfying the condition.
Therefore, option B is incorrect.
Now check for the option C that is \(t=0\) but we have already discarded this option in option B.
So, there is no need to check the option C.
Therefore, check the last option D that is the interval of time \(1\leq t\leq 3\).
The height of the ball is \(52\text{ feet}\) at time \(t=1\) as we already calculated for the option A.
Now substitute \(t=3\) in the given equation to check the interval \(1\leq t\leq 3\).
\(\begin{aligned}h(3)&=48\cdot (3)^{2}+4\\&=(48\cdot 9)+4\\&=436\end{aligned}\)
The height of the ball is \(436\text{ feet}\) at time \(t=3\) satisfies the condition completely as the height \(436\text{ feet}\) is greater than \(52\text{ feet}\).
Therefore, the boundary points of the interval \(1\leq t\leq 3\) satisfy the condition.
Now check another point which lies in the interval of time \(1\leq t\leq 3\).
So, substitute \(t=2\) in the equation \(h(t)=48t^{2}+4\).
\(\begin{aligned}h(2)&=48\cdot (2)^{2}+4\\&=(48\cdot 4)+4\\&=196\end{aligned}\)
The height of the ball is \(196\text{ feet}\) at time \(t=2\) satisfies the condition completely as the height \(196\text{ feet}\) is greater than \(52\text{ feet}\).
Therefore, the height of the ball is greater than or equal to \(52\text{ feet}\) in the interval of time \(1\leq t\leq 3\).
Thus, the correct option is \(\boxed{\bf option D}\).
Learn more:
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Answer details:
Grade: High school
Subject: Mathematics
Chapter: Quadratic equations
Keywords: Polynomial, quadratic equation, substitution, baseball, interval, height, greater than, equation, time, seconds, simplification, solution, inequality.
