Question

The number of hours of daylight varies throughout the course of a year (daylight is defined as the time between sunrise and sunset). It has been suggested to approximate the length of daylight in Maths city by the function:

() = 2.5cos( 2/365) ( − (1000 + 24 + 15+ 42/ 4 )) + 12, 0 ≤ ≤ 365,
where () is the number of daylight hours, is time measured in days ( = 0 at January 1, 2022), and radian angle measure is used.
(a) What is the lengths of daylight when = 0 and = 213?
(b) What is the shortest length of daylight? What is the value of then?
(c) What is the longest length of daylight? What is the value of then?
(d) Find the domain and range of the function using interval notation.
(e) Use the information from () − () to sketch a graph of the function, including all relevant information.
(f) Find ′′() (the second derivative of ()

Answer 1

The given function represents the approximate length of daylight in Maths City throughout the year. In order to answer the provided questions, we need to evaluate the function for specific values of time (in days) and interpret the results accordingly.

(a) To find the length of daylight when