If a stone is thrown down at 104 ft/sec from a height of 1,200 feet, its height *s* after *t* seconds is given by 

*s*(*t*) = 1,200 − 104*t* − 16*t*2,

 with *s* in feet.

(a)

Compute 

*s* *'*(*t*).

*s* *'*(*t*) = 
 
Find the stone's velocity (in ft/s) at times 

*t* = 0, 1, 2, 3,

 and 4 seconds.

*s* ′(0)= ft/s*s* ′(1)= ft/s*s* ′(2)= ft/s*s* ′(3)= ft/s*s* ′(4)= ft/s

(b)

When does the stone reach the ground, and how fast is it traveling when it hits the ground? [*HINT*: It reaches the ground when 

*s*(*t*) = 0.]

The stone is traveling at a velocity of  ft/s when it hits the ground at  seconds

We are given the height function \( s(t) \) of a stone thrown downward from a height of 1,200 feet:

\[
s(t) = 1200 - 104t - 16t^2
\]

### (a) Find \( s'(t) \), the velocity of the stone.

The derivative of \( s(t) \), denoted as \( s'(t) \), represents the velocity of the stone at any time \( t \).

We differentiate \( s(t) \) with respect to \( t \):

\[
s'(t) = \frac{d}{dt}\left(1200 - 104t - 16t^2\right)
\]

- The derivative of the constant 1200 is 0.
- The derivative of \( -104t \) is \( -104 \).
- The derivative of \( -16t^2 \) is \( -32t \).

Thus, the velocity function is:

\[
s'(t) = -104 - 32t
\]

Now, we calculate the velocity at different times:

- \( s'(0) = -104 - 32(0) = -104 \, \text{ft/s} \)
- \( s'(1) = -104 - 32(1) = -136 \, \text{ft/s} \)
- \( s'(2) = -104 - 32(2) = -168 \, \text{ft/s} \)
- \( s'(3) = -104 - 32(3) = -200 \, \text{ft/s} \)
- \( s'(4) = -104 - 32(4) = -232 \, \text{ft/s} \)

So the velocities are:

- \( s'(0) = -104 \, \text{ft/s} \)
- \( s'(1) = -136 \, \text{ft/s} \)
- \( s'(2) = -168 \, \text{ft/s} \)
- \( s'(3) = -200 \, \text{ft/s} \)
- \( s'(4) = -232 \, \text{ft/s} \)

### (b) When does the stone reach the ground, and how fast is it traveling at that point?

The stone reaches the ground when its height is 0, i.e., when \( s(t) = 0 \). We need to solve:

\[
1200 - 104t - 16t^2 = 0
\]

Rearrange the equation:

\[
16t^2 + 104t - 1200 = 0
\]

This is a quadratic equation. We can solve it using the quadratic formula:

\[
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \( 16t^2 + 104t - 1200 = 0 \), the coefficients are:

- \( a = 16 \)
- \( b = 104 \)
- \( c = -1200 \)

Now we calculate the time \( t \).The two solutions for the time are \( t = 6.0 \) seconds and \( t = -12.5 \) seconds. Since negative time doesn't make sense in this context, the stone reaches the ground after 6 seconds.

Next, we find how fast the stone is traveling when it hits the ground. We substitute \( t = 6 \) into the velocity function \( s'(t) = -104 - 32t \):

\[
s'(6) = -104 - 32(6) = -104 - 192 = -296 \, \text{ft/s}
\]

So, the stone is traveling at a velocity of \( -296 \, \text{ft/s} \) when it hits the ground.

### Summary:
- The stone reaches the ground at \( t = 6 \) seconds.
- The velocity when the stone hits the ground is \( -296 \, \text{ft/s} \).

Would you like further details or have any questions? Here are some related questions for practice:

1. What is the stone’s velocity at \( t = 5 \) seconds?
2. What would be the initial height if the stone were thrown at \( 50 \, \text{ft/s} \) instead of \( 104 \, \text{ft/s} \)?
3. How would the time to hit the ground change if the height were 1500 feet?
4. How can the equation be adjusted if the stone was thrown upward instead of downward?
5. What would the impact velocity be if the acceleration due to gravity was different?

**Tip**: When solving quadratic equations, always check the discriminant to see if real solutions exist.

If a stone is thrown down at 104 ft/sec from a height of 1,200 feet, its height s after t seconds is given by s(t) = 1,200 − 104t − 16t^2, with s in feet. (a) Compute s'(t). Find the stone's velocity (in ft/s) at times t = 0, 1, 2, 3, and 4 seconds. (b) When does the stone reach the ground, and how fast is it traveling when it hits the ground?

This problem explains how to calculate the velocity of a stone thrown downward from a height of 1,200 feet at 104 ft/sec. Learn how to differentiate to find the velocity at various times and use the quadratic formula to determine when the stone hits the ground. Suitable for high school students in Grades 10-12.

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