Chapter 7.5, Problem 65E

Hours of Daylight In Philadelphia the number of hours of daylight on day t (where t is the number of days after January 1) is modeled by the function

$L(t)=12+2.83\sin \left(\frac{2\pi }{365}(t-80)\right)$

1. (a) Which days of the year have about 10 h of daylight?
2. (b) How many days of the year have more than 10 h of daylight?

To determine

The days of the year that has about 10 hours of day light.

Answer to Problem 65E

The days of the year that has about 10 hours of day light is $\underset{¯}{{34}^{th}\text{ day}\left(\text{February 3}\right){\text{ and 308}}^{th}\text{ day}\left(\text{November 4}\right)}$.

Explanation of Solution

Given:

In Philadelphia the number of hours of daylight on day t is given by $L\left(t\right)=12+2.83\sin \left(\frac{2\pi }{365}\left(t-80\right)\right)$.

Calculation:

Use online graphing calculator and obtain the graph of the given function $10=12+2.83\sin \left(\frac{2\pi }{365}\left(t-80\right)\right)$ as shown below in Figure 1.

From Figure 1, it is observed that the required days are day 34 and day 308.

That is, the exact day is February 3^rd and November 4^th.

Therefore, the days of the year that has about 10 hours of day light is $\underset{¯}{{34}^{th}\text{ day}\left(\text{February 3}\right){\text{ and 308}}^{th}\text{ day}\left(\text{November 4}\right)}$.

To determine

The number of days of the year in which there are more than 10 hours of daylight.

Answer to Problem 65E

The number of days of the year in which there are more than 10 hours of daylight is $\underset{¯}{274\text{ days}}$.

Explanation of Solution

Use online graphing calculator and obtain the graph of the given function $10\le 12+2.83\sin \left(\frac{2\pi }{365}\left(t-80\right)\right)$ as shown below in Figure 2.

From Figure 2, it is observed that the shaded region is the days of the year in which there are more than 10 hours of daylight. That is, 308−34 = 274 days.

Therefore, the number of days of the year in which there are more than 10 hours of daylight is $\underset{¯}{274\text{ days}}$.