sam throws a ball up into the air and watches it come down. the equation for the height, h, in relation to…

sam throws a ball up into the air and watches it come down. the equation for the height, h, in relation to time, t, is h = -5t² + 25t. what is the range for the height of the ball?\n0 ≤ h ≤ 5\n0 ≤ h ≤ 25\n0 ≤ h ≤ 31.25\n2.5 < h < 31.25

sam throws a ball up into the air and watches it come down. the equation for the height, h, in relation to time, t, is h = -5t² + 25t. what is the range for the height of the ball?\n0 ≤ h ≤ 5\n0 ≤ h ≤ 25\n0 ≤ h ≤ 31.25\n2.5 < h < 31.25

Answer

Explanation:

Step1: Identify the type of function

The function $h = - 5t^{2}+25t$ is a quadratic - function in the form $y = ax^{2}+bx + c$ where $a=-5$, $b = 25$, and $c = 0$.

Step2: Find the vertex of the parabola

The $t$ - coordinate of the vertex of a quadratic function $y = ax^{2}+bx + c$ is given by $t=-\frac{b}{2a}$. Substituting $a=-5$ and $b = 25$ into the formula, we get $t=-\frac{25}{2\times(-5)}=\frac{25}{10}=2.5$.

Step3: Find the maximum value of the function

Substitute $t = 2.5$ into the function $h=-5t^{2}+25t$. Then $h=-5\times(2.5)^{2}+25\times2.5=-5\times6.25 + 62.5=-31.25+62.5 = 31.25$.

Step4: Consider the physical situation

The ball starts on the ground ($h = 0$) and then goes up and comes back down to the ground. The minimum value of $h$ is $h = 0$ (when $t = 0$ or when the ball returns to the ground). The maximum value of $h$ is $h = 31.25$. So the range of $h$ is $0\leq h\leq31.25$.

Answer:

C. $0\leq h\leq31.25$