Questions: A stone is thrown straight up from the roof of a 240-ft building. The distance (in feet) of the stone from the ground at any time t (in seconds) is given by the following.
h(t)=-16 t^2+32 t+240
When is the stone rising?
5<t<1
t<5
t<1
t>5
t>1
When is it falling?
1>t>5
t<5
t<1
t>1
t>5
If the stone were to miss the building, when would it hit the ground?
t= sec
Transcript text: A stone is thrown straight up from the roof of a 240 -ft building. The distance (in feet) of the stone from the ground at any time $t$ (in seconds) is given by the following.
\[
h(t)=-16 t^{2}+32 t+240
\]
When is the stone rising?
$55$
$t>1$
When is it falling?
$1>t>5$
$t<5$
$t<1$
$t>1$
$t>5$
If the stone were to miss the bullding, when would it hit the ground?
$t=$ $\square$ sec
Solution
Solution Steps
Step 1: Determine when the stone is rising
To find when the stone is rising, we need to determine when the velocity of the stone is positive. The velocity is the derivative of the height function h(t).
The height function is given by:
h(t)=−16t2+32t+240
The velocity v(t) is the derivative of h(t):
v(t)=dtd(−16t2+32t+240)=−32t+32
Set the velocity greater than zero to find when the stone is rising:
−32t+32>0
Solving for t:
−32t>−32t<1
Thus, the stone is rising when t<1.
Step 2: Determine when the stone is falling
The stone is falling when the velocity is negative:
−32t+32<0
Solving for t:
−32t<−32t>1
Thus, the stone is falling when t>1.
Step 3: Determine when the stone hits the ground
The stone hits the ground when h(t)=0. We solve the equation:
−16t2+32t+240=0
This is a quadratic equation of the form at2+bt+c=0, where a=−16, b=32, and c=240. We use the quadratic formula:
t=2a−b±b2−4ac
Substitute the values:
t=2(−16)−32±322−4(−16)(240)
Calculate the discriminant:
322−4(−16)(240)=1024+15360=16384
Calculate t:
t=−32−32±16384
t=−32−32±128
This gives two solutions:
t=−32−32+128=−3296=−3t=−32−32−128=−32−160=5
Since time cannot be negative, the stone hits the ground at t=5 seconds.
Final Answer
The stone is rising when t<1: t<1
The stone is falling when t>1: t>1
The stone hits the ground at t=5 seconds: t=5 sec