Explanation:
13.
the functional value (and as that the minimum and maximum) is created as a combination of constants and variables.
in our case we have 16 and k as constants, and cos(x) is variable due to the fact that x is variable.
now, 16 and k will always be 16 and k, no matter what value x has.
so, maximum and minimum only depend on cos(x) and x itself.
x has no minimum or maximum per se. it can have any value in the open interval of -infinity to +infinity (as infinity is not a value, it is only a "concept").
but cos(x) has maximum and minimum values, as that function keeps oscillating in all eternity between -1 and +1.
so, the minimum value of cos(x) is -1.
and since the minimum value of the whole function is -4, then
minimum = -4 = 16×-1 + k = -16 + k
-4 + 16 = -16 + k + 16
12 = k
and then for the maximum we get
maximum = 16×maximum(cos(x)) + 12 =
= 16×(+1) + 12 = 16 + 12 = 28
14.
as in 13. the other terms in the height function are constants.
the only driving term that determines the difference of height is the cosine function.
the maximum of the cosine function is +1.
the time between 2 maximums is the time for the cosine function to go from +1 to +1 again.
we only need to find the length of time = the minimum difference between 2 values of t, so that cos(pi×t/3) has the same value of +1.
so, what is the period of cos(x) ?
e.g.
cos(x) = 1 for x = 0.
so, what is the closest x-value, for which cos(x) is again 1 ?
is it pi ? no. cos(pi) = -cos(0) = -1
but 2pi ? yes. cos(2pi) = cos(0) = +1
so, for what values of t do we get a difference of 2pi in the argument of cos(pi×t/3) ?
pi×(t + x)/3 - pi×t/3 = 2pi
pi×(t + x) - pi×t = 6pi
pi×t + pi×x - pi×t = 6pi
pi×x = 6pi
x = 6
so, the functional value for t+6 is the same as for t.
that means it takes the pendulum 6 seconds to go from maximum height to maximum height again.