If a ball is thrown straight up into the air with an initial velocity of 95 ft/s, its height in feet after t seconds is given by

f(t)=95t−16t^2

Find the average velocity for the time period beginning when t=1 and lasting
(i) 0.5 seconds

(ii) 0.1 seconds

(iii) 0.01 seconds

(iii) 0.001 seconds

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Well, let's calculate the average velocities for each time period and see what we get!

(i) For the time period of 0.5 seconds, we can plug in the values into the equation f(t) = 95t - 16t^2. So, f(1) = 95(1) - 16(1)^2 = 95 - 16 = 79. The height at t=1 is 79 ft. Now, let's calculate the height at t=1.5. f(1.5) = 95(1.5) - 16(1.5)^2 = 142.5 - 36 = 106.5.

The average velocity is the change in height divided by the change in time. So, the average velocity for the time period of 0.5 seconds is (106.5 - 79) / (1.5 - 1) = 27.5 / 0.5 = 55 ft/s.

(ii) For the time period of 0.1 seconds, we can again use the equation f(t) = 95t - 16t^2. So, f(1) = 79 ft as we calculated before. Now, let's calculate the height at t=1.1. f(1.1) = 95(1.1) - 16(1.1)^2 = 104.5 - 19.36 = 85.14.

The average velocity is the change in height divided by the change in time. So, the average velocity for the time period of 0.1 seconds is (85.14 - 79) / (1.1 - 1) = 6.14 / 0.1 = 61.4 ft/s.

(iii) For the time period of 0.01 seconds, we can use the equation f(t) = 95t - 16t^2. Again, f(1) = 79 ft. Now, let's calculate the height at t=1.01. f(1.01) = 95(1.01) - 16(1.01)^2 = 80.95 - 16.3216 = 64.6284.

The average velocity is the change in height divided by the change in time. So, the average velocity for the time period of 0.01 seconds is (64.6284 - 79) / (1.01 - 1) = -14.3716 / 0.01 = -1437.16 ft/s. Oh no, that's a negative velocity! Looks like the ball is heading downwards pretty fast here!

(iii) For the time period of 0.001 seconds, using the equation f(t) = 95t - 16t^2, we have f(1) = 79 ft. Now, let's calculate the height at t=1.001. f(1.001) = 95(1.001) - 16(1.001)^2 = 79.595 - 16.032032 = 63.562968.

The average velocity is the change in height divided by the change in time. So, the average velocity for the time period of 0.001 seconds is (63.562968 - 79) / (1.001 - 1) = -15.437032 / 0.001 = -15437.032 ft/s. Whoa, that's a really fast downward velocity! I hope no one is standing underneath the ball.

So there you have it! The average velocities for the time periods are:

(i) 55 ft/s,
(ii) 61.4 ft/s,
(iii) -1437.16 ft/s, and
(iii) -15437.032 ft/s.

Remember, these are just the average velocities for the specific time periods. But hey, at least we got a good mix of positive and negative velocities, right?

To find the average velocity for a given time period, we need to calculate the displacement and divide it by the time interval. The displacement can be found by subtracting the initial height from the final height.

(i) For a time period of 0.5 seconds,
Initial height (t=1) = f(1) = 95(1) - 16(1)^2
Final height (t=1.5) = f(1.5) = 95(1.5) - 16(1.5)^2
Displacement = Final height - Initial height = f(1.5) - f(1)
Average velocity = Displacement / Time interval = (f(1.5) - f(1)) / 0.5

(ii) For a time period of 0.1 seconds,
Initial height (t=1) = f(1) = 95(1) - 16(1)^2
Final height (t=1.1) = f(1.1) = 95(1.1) - 16(1.1)^2
Displacement = Final height - Initial height = f(1.1) - f(1)
Average velocity = Displacement / Time interval = (f(1.1) - f(1)) / 0.1

(iii) For a time period of 0.01 seconds,
Initial height (t=1) = f(1) = 95(1) - 16(1)^2
Final height (t=1.01) = f(1.01) = 95(1.01) - 16(1.01)^2
Displacement = Final height - Initial height = f(1.01) - f(1)
Average velocity = Displacement / Time interval = (f(1.01) - f(1)) / 0.01

(iv) For a time period of 0.001 seconds,
Initial height (t=1) = f(1) = 95(1) - 16(1)^2
Final height (t=1.001) = f(1.001) = 95(1.001) - 16(1.001)^2
Displacement = Final height - Initial height = f(1.001) - f(1)
Average velocity = Displacement / Time interval = (f(1.001) - f(1)) / 0.001

To find the specific values, you can substitute the respective values of t into the function f(t) and perform the calculations.

To find the average velocity for a given time period, you need to calculate the change in height divided by the change in time. Let's calculate the average velocity for each time period.

(i) For the time period of 0.5 seconds:
To find the change in height, we need to plug in t = 1.5 and t = 1 into the equation f(t) = 95t - 16t^2 and subtract the two results:
Change in height = f(1.5) - f(1)

Substituting t = 1.5:
f(1.5) = 95(1.5) - 16(1.5^2)

Substituting t = 1:
f(1) = 95(1) - 16(1^2)

Calculate both of these to find the change in height.

Next, calculate the change in time:
Change in time = 0.5 seconds

Finally, divide the change in height by the change in time to find the average velocity.

(ii) For the time period of 0.1 seconds:
Follow the same steps as above, but now calculate the change in height and change in time using t = 1.1 and t = 1.

(iii) For the time period of 0.01 seconds:
Follow the same steps as above, but now calculate the change in height and change in time using t = 1.01 and t = 1.

(iv) For the time period of 0.001 seconds:
Follow the same steps as above, but now calculate the change in height and change in time using t = 1.001 and t = 1.

By following these steps, you can find the average velocity for each time period.