(1) A toy truck travels at a rate of 0.5t ft/sec after t seconds of travel (so after 6 seconds the truck is traveling 3 ft/sec).
a) Graph the speed of the truck vs. time for 0<t<10.
b) Find the distance the truck has traveled in the first t seconds for 0<t<10. Call it T(t).
c) Find T'(5)
(2) A micorgram of Magnesium 28 contains approximately 3*10^16 atoms and this isotope has a half life of 21 hours.
a) Express the number of atoms of Magnesium 28 as a function of t, time in hours.
b) Approximately how many atoms of Magnesium 28 decay in the one second after 5 hours have passed? (Hint: what is the matematical relation for the decay as related to the amount?)
“(1) A toy truck travels at a rate of 0.5t ft/sec after t seconds of travel (so after 6 seconds the truck is traveling 3 ft/sec). “
why is it going suddenly slower, you said it was going at .5 ft/sec.
This is really a physics problem poorly expressed. I suspect it is accelerating.
v = Vo + a t
Vo = 0
v = a t = .5 t
v versus t is a straight line through the origin with slope 1/2
T(t) = (1/2) a t^2
= .25 t^2
quadratic with slope zero at origin and going through (10, 25)
I assume T' is the time derivative of T, or the speed
T'(5) = v(5) = 2.5 ft/s
starts with 3 * 10^16 atoms
Meaning of half life
for an exponential decay in time, some constant k
mass = initial mass e^-kt
or
M = Mo e^-kt
now when is m = .5 Mo ?
.5 = e^-k T where T is half life
ln .5 = -k T
ln .5 = -k (21 hr)
-.6931 = - 21*k
k = .033
So for this stuff
M = Mo e^-.033 t where t is in hours
here Mo is 3*10^16 so make your graph
Part b
we want d M/dt in seconds when t = 5 hr
dM / dt = -(3*10^16).033 e^- (.033*5)
you can do that
then that is number lost per hour
divide by 3600 to get per second.
(1)
a) To graph the speed of the truck vs. time, we can use the given information that the speed of the truck is given by the expression 0.5t ft/sec after t seconds. We can create a table of values for different values of t and calculate the corresponding speeds. Let's create a table for t values from 0 to 10:
| t (seconds) | Speed (ft/sec) |
|------------|----------------|
| 0 | 0 |
| 1 | 0.5 |
| 2 | 1.0 |
| 3 | 1.5 |
| 4 | 2.0 |
| 5 | 2.5 |
| 6 | 3.0 |
| 7 | 3.5 |
| 8 | 4.0 |
| 9 | 4.5 |
| 10 | 5.0 |
Using these values, we can plot a graph with time on the x-axis and speed on the y-axis. Connect the points on the graph to get a smooth curve.
b) To find the distance the truck has traveled in the first t seconds, we need to integrate the speed function with respect to time. Integrating the given speed function 0.5t with respect to t will give us the distance function T(t). Let's calculate this integral:
∫(0.5t) dt = 0.25t^2 + C
where C is the constant of integration.
Since we are interested in finding the distance traveled in the first t seconds, we substitute t into the distance function:
T(t) = 0.25t^2 + C
Now we need to find the value of C. Since the truck starts from rest (speed = 0) at t = 0, we can substitute these values into the distance function to solve for C:
0 = 0.25(0^2) + C
C = 0
So the distance function for the first t seconds is:
T(t) = 0.25t^2
c) To find T'(5), we need to differentiate the distance function T(t) with respect to t. Let's calculate the derivative:
T'(t) = d/dt (0.25t^2) = 0.5t
Substituting t = 5 into the derivative function:
T'(5) = 0.5(5)
T'(5) = 2.5 ft/sec
Therefore, T'(5) = 2.5 ft/sec.
(2)
a) To express the number of atoms of Magnesium 28 as a function of time, we need to consider the decay of the atoms over time. The number of atoms at a given time is given by the initial number of atoms multiplied by 0.5^(t/h), where t is the time in hours and h is the half-life of the isotope (21 hours in this case). So the function can be expressed as:
N(t) = Initial number of atoms * 0.5^(t/21)
b) To estimate the number of atoms of Magnesium 28 that decayed in the one second after 5 hours have passed, we need to find the difference between the number of atoms at 5 hours and the number of atoms at 5 hours and 1 second.
Using the function N(t) from part a), we can calculate:
N(5) = Initial number of atoms * 0.5^(5/21)
N(5+1) = Initial number of atoms * 0.5^((5+1)/21)
The approximately number of atoms that decayed in one second after 5 hours can be calculated as:
Approximate decayed atoms = N(5) - N(5+1)
Substituting the values and evaluating the expression will give the approximate number of atoms decayed.