Question

a projectile is fired into the air from the ground with an initial velocity of 80 feet per second. the function $h(t)=-16t^{2}+80t$ 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 $- 16t^{2}+80t = 0$. Factor out $-16t$: $-16t(t - 5)=0$. Solving gives $t = 0$ (launch - time) and $t = 5$ (landing - time). The domain represents time, and time starts at $t = 0$ and ends when it hits the ground.

Step2: Find the vertex of the parabola

The function $h(t)=-16t^{2}+80t$ is a quadratic function in the form $y = ax^{2}+bx + c$ with $a=-16$, $b = 80$, $c = 0$. The $t$ - coordinate of the vertex is $t=-\frac{b}{2a}=-\frac{80}{2\times(-16)}=\frac{80}{32}=2.5$. Substitute $t = 2.5$ into $h(t)$: $h(2.5)=-16\times(2.5)^{2}+80\times2.5=-16\times6.25 + 200=-100 + 200 = 100$. The range represents height, and the maximum height is 100 and height starts at 0.

Answer:

Domain: $0\leq t\leq5.00$
Range: $0\leq h\leq100.00$