27.3k views
5 votes
13. The function y=16cos(x)+k has a minimum value of −4. What is its maximum value? Explain. 14. The height a swinging pendulum can be modeled by the function h(t)=4.5cos(πt/3)+14, where t is the amount of time, in seconds, since the pendulum was released. How many seconds pass between two consecutive times where the pendulum is at its maximum height?

User Kai Wang
by
8.9k points

1 Answer

1 vote

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.

User Tim Pigden
by
8.8k points