Question

a projectile is fired into the air from the ground with an initial velocity of 240 feet per second. the function h(t)=-16t² + 240t can be used to determine the height, h, of the projectile at t seconds. determine the practical domain and range of the function using a graphing calculator. (round your answers to 2 decimals if necessary) domain: range: question help: message instructor submit question

Explanation:

Step1: Find when the projectile hits the ground

Set $h(t)=0$. So, $- 16t^{2}+240t = 0$. Factor out $-16t$: $-16t(t - 15)=0$. The solutions are $t = 0$ and $t=15$. The time $t$ starts at $t = 0$ (when it's fired) and ends when it hits the ground at $t = 15$. The domain represents the possible values of $t$.

Step2: Find the vertex of the parabola

The function $h(t)=-16t^{2}+240t$ is a quadratic function in the form $y = ax^{2}+bx + c$ where $a=-16$, $b = 240$, $c = 0$. The $t$-coordinate of the vertex is $t=-\frac{b}{2a}=-\frac{240}{2\times(-16)}=\frac{240}{32}=7.5$.

Step3: Find the maximum height

Substitute $t = 7.5$ into $h(t)$: $h(7.5)=-16\times(7.5)^{2}+240\times7.5=-16\times56.25 + 1800=-900+1800 = 900$. The range represents the possible values of $h(t)$. Since the parabola opens downwards ($a=-16<0$), the maximum value of $h(t)$ is 900 and it starts at $h(t)=0$.

Answer:

Domain: $0\leq t\leq15.00$
Range: $0\leq h(t)\leq900.00$