a stone is thrown striaght up from the roof of an 80 foot building. the height of the stone above the ground t seconds later is giving by the function

h(t)=-16t^2 + 64t +80
during what time is the stone rising?
when is the stone falling?
what is the maximum upward velocity?
when does it reach its maximum height above the ground?

h '(t) = -32t + 64

the stone is rising when h '(t) is positive
-32t + 64 ≥ 0
-32t > -64
t < 2

similarly the stone is falling for t > 2

the velocity is -32t + 64
clearly the max upwards velocity occurs when t=0 (at the beginning)
and it is 0 + 64 or 64 ft/sec

max height is when -32t + 64 = 0
t = 2
so h(2) = -64 + 128 + 80
= 144 ft

Well, well, we have a stone trying to reach for the stars! Let's see how high it can go and when it comes crashing back down.

To determine when the stone is rising, we need to find the intervals where the height function h(t) is increasing. To do that, we need to look at the derivative.

The derivative of h(t) is given by h'(t) = -32t + 64.

When this derivative is positive, the stone is rising. So let's solve -32t + 64 > 0.

Brace yourselves, math is coming!

-32t + 64 > 0
-32t > -64
t < 2

Therefore, the stone is rising when t < 2.

Next up, when is the stone falling? The stone falls when h'(t) is negative. Solving for -32t + 64 < 0 gives us:

-32t + 64 < 0
-32t < -64
t > 2

Uh-oh, it seems the stone has reached its peak and started falling after t = 2.

To find the maximum upward velocity, we can look at the vertex of the parabolic function. The x-coordinate of the vertex is given by -b/(2a), where a = -16 and b = 64.

t = -64 / (2 * -16)
t = 2

Plug this back into h'(t), and we find the maximum upward velocity:

h'(2) = -32 * 2 + 64
h'(2) = 0 ft/s

It appears the maximum upward velocity is 0 ft/s. That's quite a stall, stone!

Lastly, we need to determine when the stone reaches its maximum height. To do that, we look at the vertex of the parabola given by h(t). The x-coordinate of the vertex is again t = 2.

So at t = 2, the stone reaches its maximum height above the ground.

Hope that answers your questions! Keep reaching for the stars, or at least watch out for stones falling from the roof of buildings!

To determine when the stone is rising or falling, we need to analyze the sign of the velocity of the stone. The velocity function can be found by taking the derivative of the height function, h(t).

1. Find the derivative of h(t) to get the velocity function, v(t):
h'(t) = -32t + 64
v(t) = -32t + 64

2. When the stone is rising, the velocity (v(t)) is positive. To find the time during which the stone is rising, set v(t) > 0 and solve for t:
-32t + 64 > 0
-32t > -64
t < 2

The stone is rising for t < 2 seconds.

3. When the stone is falling, the velocity (v(t)) is negative. To find the time during which the stone is falling, set v(t) < 0 and solve for t:
-32t + 64 < 0
-32t < -64
t > 2

The stone is falling for t > 2 seconds.

4. To find the maximum upward velocity, we need to find the vertex of the parabolic function h(t). The maximum/minimum point occurs at the vertex of the parabola.

To find the vertex of the parabolic function h(t), use the formula:
t = -b / (2a)

For h(t) = -16t^2 + 64t + 80:
a = -16, b = 64

t = -64 / (2 * -16)
t = -64 / -32
t = 2

Substituting t = 2 into the velocity function v(t) = -32t + 64:
v(2) = -32(2) + 64
v(2) = -64 + 64
v(2) = 0

The maximum upward velocity is 0 ft/s.

5. The stone reaches its maximum height when the velocity becomes 0. The time at which this occurs is given by the same value of t we found in step 4.

The stone reaches its maximum height above the ground at t = 2 seconds.

To determine when the stone is rising or falling, we need to look at the velocity of the stone. The velocity can be obtained by taking the derivative of the height function with respect to time.

1. Finding the velocity function:
First, let's find the derivative of the height function, h(t), with respect to time, t.

h'(t) = -32t + 64

Now, we can analyze the sign of the velocity to determine when the stone is rising or falling.

2. Determining when the stone is rising or falling:
The stone is rising when the velocity is positive (greater than 0) and falling when the velocity is negative (less than 0). We can solve the velocity function to find the time intervals.

-32t + 64 > 0
-32t > -64
t < 2

From this, we can see that the stone is rising during the time interval 0 < t < 2.

3. Finding the maximum upward velocity:
The maximum upward velocity occurs at the highest point of the stone's trajectory, which is when the velocity is zero.

Setting h'(t) = 0:
-32t + 64 = 0
t = 2

Therefore, the maximum upward velocity occurs at t = 2 seconds.

4. Finding the time at which the stone reaches its maximum height above the ground:
To find the time at which the stone reaches its maximum height above the ground, we can find the maximum point of the height function. This occurs at the vertex of the parabolic function.

The vertex of a parabolic function in the form h(t) = at^2 + bt + c can be found using the formula:

t = -b / (2a)

For our function h(t) = -16t^2 + 64t + 80:

t = -64 / (2*(-16))
t = -64 / (-32)
t = 2

Therefore, the stone reaches its maximum height above the ground at t = 2 seconds.