Question

The function f(t) = 65 sin (pi over 5t) + 35 models the temperature of a periodic chemical reaction where t represents time in hours. What are the maximum and minimum temperatures of the reaction, and how long does the entire cycle take?

Answer 1

Maximum: 100°; minimum: −30°; period: 10 hours
sin(πt/5) varies between -1 and 1, so f(t) varies between -30 and 100. P=2π/π/5=10

Answer 2

1. Minimum and maximum temperatures are
   `-30,100`
2. Time taken to complete one complete cycle is 10 hours.

Explanation:

The general sinusoidal periodic function is,

`f(t)=a\sin (b(t+c))+d`

here,

1. amplitude = a,
2. period =
   `(2\pi)/(b)`,
3. horizontal or phase shift = c,
4. vertical shift = d,

Comparing this with the given function
`f(t)=65\sin(\pi)/(5)x+35`, we get

1. amplitude = a = 65,
2. period =
   `(2\pi)/((\pi)/(5))` = 10,
3. vertical shift = d = 35, so the axis of symmetry will be,
   `y=35`

The maximum and minimum temperatures will be,

`=35\pm 65`

`=-30,100`

Time taken to complete one complete cycle is the period, so it is 10 hours.