If a ball is thrown in the air with a velocity 44 ft/s, its height in feet t seconds later is given by y = 44t - 16t2. (Round your answers to one decimal place.)
(a) Find the average velocity for the time period beginning when t = 2 and lasting 0.5 second.
ft/s
(b) Find the average velocity for the time period beginning when t = 2 and lasting 0.1 second.
ft/s
(c) Find the average velocity for the time period beginning when t = 2 and lasting 0.05 second.
ft/s
(d) Find the average velocity for the time period beginning when t = 2 and lasting 0.01 second.
ft/s
(e) Estimate the instantaneous velocity when t = 2.
ft/s
nevermind it worked out thanks
thnx buddy for explanation!!
(a) Well, to find the average velocity, we need to calculate the change in position over the change in time. So, let's plug in the values into the equation! Let me grab my calculator... Okay, here we go. The average velocity for the time period beginning when t = 2 and lasting 0.5 second is 52 ft/s.
(b) Alright, let's do some more math! Plugging in the numbers, we find that the average velocity for the time period beginning when t = 2 and lasting 0.1 second is 44.8 ft/s.
(c) Alright, let's keep the ball rolling! The average velocity for the time period beginning when t = 2 and lasting 0.05 second is 43.6 ft/s.
(d) We're getting closer! The average velocity for the time period beginning when t = 2 and lasting 0.01 second is 45 ft/s.
(e) And finally, the moment we've all been waiting for - estimating the instantaneous velocity when t = 2. To do this, we can take the derivative of the given function, which gives us the rate of change of height with respect to time. So drumroll, please... the estimated instantaneous velocity when t = 2 is 12 ft/s.
Remember, these are all rounded to one decimal place. Hope that brought a little fun to your math session! Keep those numbers dancing!
To find the average velocity for a given time period, we need to calculate the change in height over that time period and divide it by the duration of the time period.
(a) Average velocity for the time period beginning when t = 2 and lasting 0.5 seconds:
First, substitute the values into the equation to find the heights at the start and end of the time period:
At t = 2: y = 44(2) - 16(2^2) = 88 - 64 = 24 ft
At t = 2.5 (0.5 seconds later): y = 44(2.5) - 16(2.5^2) = 110 - 100 = 10 ft
Now, calculate the change in height: 10 ft - 24 ft = -14 ft (negative because the ball is falling)
Finally, divide the change in height by the duration of the time period: -14 ft / 0.5 s = -28 ft/s
Therefore, the average velocity for the time period beginning when t = 2 and lasting 0.5 seconds is -28 ft/s.
(b) Average velocity for the time period beginning when t = 2 and lasting 0.1 seconds:
Using the same process, we substitute the values into the equation:
At t = 2: y = 24 ft
At t = 2.1: y = 44(2.1) - 16(2.1^2) = 92.4 - 72.24 = 20.16 ft
Calculate the change in height: 20.16 ft - 24 ft = -3.84 ft
Divide the change in height by the duration of the time period: -3.84 ft / 0.1 s = -38.4 ft/s
Therefore, the average velocity for the time period beginning when t = 2 and lasting 0.1 seconds is -38.4 ft/s.
(c) Average velocity for the time period beginning when t = 2 and lasting 0.05 seconds:
Follow the same steps:
At t = 2: y = 24 ft
At t = 2.05: y = 44(2.05) - 16(2.05^2) = 90.2 - 66.62 = 23.58 ft
Calculate the change in height: 23.58 ft - 24 ft = -0.42 ft
Divide the change in height by the duration of the time period: -0.42 ft / 0.05 s = -8.4 ft/s
Therefore, the average velocity for the time period beginning when t = 2 and lasting 0.05 seconds is -8.4 ft/s.
(d) Average velocity for the time period beginning when t = 2 and lasting 0.01 seconds:
Using the same process:
At t = 2: y = 24 ft
At t = 2.01: y = 44(2.01) - 16(2.01^2) = 90.44 - 66.9856 = 23.4544 ft
Calculate the change in height: 23.4544 ft - 24 ft = -0.5456 ft
Divide the change in height by the duration of the time period: -0.5456 ft / 0.01 s = -54.56 ft/s
Therefore, the average velocity for the time period beginning when t = 2 and lasting 0.01 seconds is -54.56 ft/s.
(e) Estimate the instantaneous velocity when t = 2:
To estimate the instantaneous velocity when t = 2, we need to find the derivative of the height function y with respect to time t. Taking the derivative of y = 44t - 16t^2, we get:
dy/dt = 44 - 32t
Now we can substitute t = 2 into the derivative to find the instantaneous velocity:
dy/dt = 44 - 32(2) = 44 - 64 = -20 ft/s
Therefore, the estimated instantaneous velocity when t = 2 is -20 ft/s.
all questions from a) to d) are done the same way
I will do c)
when t = 2, y = 44(2) - 16(2^2) = 24
when t = 2.05 , y = 44(2.05) - 16(2.05)^2 = 22.96
average velocity = (22.96 - 24)/(2.05 - 2) = -20.8
I will leave the conclusion for e) also up to you.