Question

a ball is thrown straight up from the top of a 112 - foot tall building with an initial speed of 96 feet per second. the height of the ball as a function of time can be modeled by the function (h(t)=-16t^{2}+96t + 112). a) graph the equation in desmos. sketch the graph on your work - page. label the vertex and zeros! b) how long will it take for the ball to hit the ground? seconds b) what is the maximum height of the ball? 256 feet c) will the ball ever reach a height of 300 feet? yes or no no explain why on your work paper.

Explanation:

Step1: Identify the function

The height - time function is $h(t)=-16t^{2}+96t + 112$, which is a quadratic function in the form $y = ax^{2}+bx + c$ where $a=-16$, $b = 96$, and $c = 112$.

Step2: Find the vertex of the parabola (for maximum height)

The $t$ - coordinate of the vertex of a quadratic function $y = ax^{2}+bx + c$ is given by $t=-\frac{b}{2a}$. Substituting $a=-16$ and $b = 96$ into the formula, we get $t=-\frac{96}{2\times(-16)}=\frac{-96}{-32}=3$.
Then substitute $t = 3$ into the function $h(t)$: $h(3)=-16\times3^{2}+96\times3 + 112=-16\times9 + 288+112=-144 + 288+112=256$ feet.

Step3: Find the time when the ball hits the ground

Set $h(t)=0$, so $-16t^{2}+96t + 112 = 0$. Divide the entire equation by - 16 to simplify: $t^{2}-6t - 7=0$. Factor the quadratic equation: $(t - 7)(t+1)=0$. Then $t = 7$ or $t=-1$. Since time $t\geq0$, the ball hits the ground at $t = 7$ seconds.

Step4: Check if the ball reaches 300 feet

Set $h(t)=300$, so $-16t^{2}+96t + 112 = 300$. Rearrange to get $-16t^{2}+96t - 188 = 0$. Divide by - 4: $4t^{2}-24t + 47 = 0$. Calculate the discriminant $\Delta=b^{2}-4ac$, where $a = 4$, $b=-24$, and $c = 47$. $\Delta=(-24)^{2}-4\times4\times47=576 - 752=-176<0$. So the ball does not reach a height of 300 feet.

Answer:

a) Graphing in Desmos and sketching is a visual - based task which can be done by inputting the function $y=-16x^{2}+96x + 112$. The vertex is at $(3,256)$ and the zeros are at $x=-1$ and $x = 7$ (we discard $x=-1$ as time cannot be negative in this context).
b) The maximum height of the ball is 256 feet.
b) It takes 7 seconds for the ball to hit the ground.
c) no