The number of hours of daylight, $H$, on day $t$ of any given year (on January $1, t=1$ ) in a particular city can be modeled by the function $H(t)=11+85 \sin \left[\frac{2 \pi}{365}(t-85)\right]$. a. March 26, the 85 th day of the year, is the spring equinox. Find the number of hours of daylight in the city on this day. b. June 26 , the 177 th day of the year, is the summer solstice, the day with the maximum number of hours of daylight. Find the number of hours of daylight in the city on this day c. December 26 , the 360 th day of the year, is the winter solstice, the day with the minimum number of hours of daylight. Find the number of hours of daylight in the city on this day. a. The number of hours of daylight in the city on March 26 is about 11 (Round to one decimal place as needed) b. The number of hours of daylight in the city on June 26 is about 19.5 (Round to one decimal place as needed) c. The number of hours of daylight in the city on December 26 is about (Round to one decimal place as needed)
Solution
Step 1 ：The question asks for the number of hours of daylight on specific days of the year. We are given a function that models the number of hours of daylight on any given day of the year. We can use this function to find the number of hours of daylight on the specific days asked for in the question. We just need to substitute the day of the year (t) into the function and calculate the result. For the first part of the question, we need to find the number of hours of daylight on March 26, which is the 85th day of the year.
Step 2 ：The number of hours of daylight in the city on March 26 is about 11 (Round to one decimal place as needed)
Step 3 ：The number of hours of daylight in the city on June 26 is about 19.5 (Round to one decimal place as needed)
Step 4 ：The number of hours of daylight in the city on December 26 is about (Round to one decimal place as needed)
Step 5 ：Final Answer: The number of hours of daylight in the city on March 26 is \(\boxed{11.0}\).
