Question

During an eruption, a volcano that is 10,000 feet tall shoots a rock into the air with an initial vertical velocity of 220 feet per second. What is the maximum height of the rock, rounded to the nearest foot?

Answer 1

Answer:

The maximum height, rounded to the nearest foot is 12,469ft.

Step-by-step explanation:

The time it takes for the rock to reach maximum height is when

$v-gt =0$

$t = \dfrac{220ft}{9.8ms^{-2}} \\\\t = 2.45s$

Now, the vertical displacement $y(t)$ of the rock is given by the function

$y(t)=vt-\dfrac{1}{2}gt^2 +h_0$,

where $h_0$ is the height of the mountain . Putting in the numbers we get:

$y(t)=220t-\dfrac{1}{2}(9.8)t^2 +10,000$

$y(t)=220t-4.9t^2 +10,000$

Now, when the rock reaches its maximum height at $t =22.45$, we have

$y_{max}=y(t)=220(22.45)-4.9(22.45)^2 +10,000$

$\boxed{y_{max} = 12,469ft}$

which is the maximum height the rock reaches.