(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) To solve parts (a), (b), and (c), we need to find a function for the speed and distance traveled by the truck at any given time.
a) Graphing the speed vs. time:
Since the speed of the truck is given as 0.5t ft/sec, we can plot the speed on the y-axis and time on the x-axis. The equation of the speed function is:
Speed = 0.5t ft/sec
Plotting the graph will give a linear line that passes through the origin (0, 0), and the slope of the line represents the speed of the truck. So, start plotting points by substituting different values of t into the equation, such as t = 1, 2, 3, etc., up to t = 10.
b) Finding the distance traveled:
To find the distance traveled by the truck at any given time, we need to integrate the speed function over the interval [0, t]:
T(t) = ∫(0 to t) 0.5t dt
To integrate, we can use the power rule of integration:
T(t) = (0.5/2)t^2 = 0.25t^2 (ft)
So, the function for the distance traveled T(t) is T(t) = 0.25t^2 ft.
c) Finding T'(5):
To find the derivative (or rate of change) of the distance function T(t), we can differentiate T(t) with respect to t:
T'(t) = 0.25 * d/dt (t^2) = 0.5t
To find T'(5), simply substitute t = 5 into the derived function:
T'(5) = 0.5 * 5 = 2.5 ft/sec
Therefore, T'(5) = 2.5 ft/sec.
(2) To solve parts (a) and (b), we need to find the number of atoms of Magnesium 28 as a function of time and calculate the decay rate.
a) Expressing the number of atoms as a function of time:
The number of atoms of Magnesium 28 at any given time can be modeled by the equation:
N(t) = N0 * (0.5)^(t / T)
Where N0 is the initial number of atoms, t is the time in hours, and T is the half-life of Magnesium 28.
b) Calculating the number of decaying atoms:
To calculate the number of atoms of Magnesium 28 that decay in the one second after 5 hours have passed, we can subtract the number of atoms at 5 hours from the number of atoms at 5 hours and 1 second:
Decaying atoms = N(5) - N(5 + 1/3600)
Substitute the values into the equation:
Decaying atoms = N0 * (0.5)^(5 / T) - N0 * (0.5)^(5.000028 / T)
Calculate the value using the values given in the problem.