Question

a ball is thrown straight up from the top of an 80 - foot tall building with an initial speed of 64 feet per second. the height of the ball as a function of time can be modeled by the function h(t)=-16t^2 + 64t + 80. a) what is the maximum height of the ball? height = feet. b) how long is the ball in the air? time = seconds. c) at what time will the ball reach 128 feet? time = seconds and seconds. d) will the ball ever reach a height of 150 feet? yes or no?

Explanation:

Step1: Identify the quadratic - function form

The height function of the ball is $h(t)=-16t^{2}+64t + 80$, which is in the form of a quadratic function $y = ax^{2}+bx + c$ where $a=-16$, $b = 64$, and $c = 80$.

Step2: Find the maximum - height (a)

The $t$ - value of the vertex of a quadratic function $y = ax^{2}+bx + c$ is given by $t=-\frac{b}{2a}$. Substituting $a=-16$ and $b = 64$ into the formula, we get $t=-\frac{64}{2\times(-16)}=2$. Then substitute $t = 2$ into the height - function $h(t)$: $h(2)=-16\times2^{2}+64\times2 + 80=-16\times4 + 128+80=-64 + 128+80=144$ feet.

Step3: Find the time the ball is in the air (b)

The ball hits the ground when $h(t)=0$. So we set $-16t^{2}+64t + 80 = 0$. Divide the entire equation by - 16 to simplify: $t^{2}-4t - 5=0$. Factor the quadratic equation: $(t - 5)(t+1)=0$. Then $t = 5$ or $t=-1$. Since time cannot be negative, the ball is in the air for $t = 5$ seconds.

Step4: Find the time when the ball reaches 128 feet (c)

Set $h(t)=128$, so $-16t^{2}+64t + 80 = 128$. Rearrange it to the standard quadratic - form: $-16t^{2}+64t - 48 = 0$. Divide by - 16: $t^{2}-4t + 3=0$. Factor the quadratic equation: $(t - 1)(t - 3)=0$. So $t = 1$ second and $t = 3$ seconds.

Step5: Check if the ball reaches 150 feet (d)

Set $h(t)=150$, so $-16t^{2}+64t + 80 = 150$. Rearrange to $-16t^{2}+64t - 70 = 0$ or $8t^{2}-32t + 35 = 0$. Calculate the discriminant $\Delta=b^{2}-4ac$, where $a = 8$, $b=-32$, and $c = 35$. $\Delta=(-32)^{2}-4\times8\times35=1024 - 1120=-96\lt0$. So the ball does not reach a height of 150 feet, and the answer is no.

Answer:

a) height = 144 feet
b) time = 5 seconds
c) time = 1 second and 3 seconds
d) no