CPhill

Let d be the distance  to the water

The time to fall d distance is given by :

-d [meters] = (-4.9m/s^2)t^2

Rearranging this  and solving for t  we have

sqrt(d/4.9)  = t

And, the speed of sound  at 20C ≈ 343m/s

So......the time for the sound to travel from the water to the observer =

  [d meters] / 343m/s  =

d/343

So the total time is given by :

Time to fall + Time for sound return  = 6.58 sec

sqrt(d/4.9)  + d/343  =  6.58

Solving this for d, we have

d  ≈  179.712m

We need to solve for the time to hit the ground...and we have

0  = -4.9t^2 + 15  subtract 15 from both sides

-15 = -4.9 t^2    divide both sides by -4.9

-15 / - 4.9  = t^2       take the square root of both sides

t = about 1.75s

So......the soccer ball landed

12.3 (1.75)  = about 21.525m from the base of the cliff

Yep....that one is good, too....

How about this one, DS???

Yes, prime.....!!!

We need to find the time the ball traveled

We can use this equation :

0 = 3.20sin(55)t  - 4.9t^2   divide through by t

0 = 3.20sin(55)  - 4.9t    add 4.9t to both sides

4.9t = 3.20sin(55)    divide both sides by 4.9

t = 3.20sin(55)/ 4.9  = about .535s

So.....the horizontal displacement  is given by :

d  = 3.2cos(55)(.535) = about  0.982 m

I believe we can use similar triangles to solve this

In one second the current rate is 29/36m/s   and the swimmer's rate is 20/36m/s...so...

29/20  = d/500   where d is the distance downstream ....multiply both sides by 500

500*29/20 = d =  725m

Vf = Vi + (-9.8m/s^2)*t  whereVf is the velocity at any time, Vi = 16.67m/s and t = 1  ......so we have

Vf = 16.67/m/s + (-9.8m/s^2) (1s)  =

Vf = 16.67m/s - 9.8m/s  =

Vf = 6.87m/s   [ the potato is still rising after 1 sec ]

Vf  = Vi  + (-9.8m/s^2) t           where Vf is the final velocity, Vi is the initial velcocity, -9.8m/s^2  is the acceleration due to gravity, and t is the time.......note......Vi  = 0, since the penny is dropped....so we have

Vf = -9.8 (4.5)  ≈  -44.1 m/s    [ we need the negative because the object is moving in a downward direction ]

sin(3x - 12)  / (x - 4)     as x →  4

We have a  0/0   situation, here....so we can use L'Hospital's Rule

Taking the derivative with respect to x on top/bottom, we have

3cos(3x - 12)      as  x  →  4  =

3cos(3(4) - 12)  =

3cos(12 - 12)  =

3cos(0)   =

3 * 1  =

3

This graph confirms the results :   https://www.desmos.com/calculator/60evrkvzkx