Chapter 13.5, Problem 21CYU

a.

To determine

To find: on what days will Hartford have exactly $10\frac{1}{2}$ hours of daylight.

Answer to Problem 21CYU

October 20^th, February 19^th.

Explanation of Solution

Given information: The number of hours of daylight d in Hartford, Connecticut, may be

approximated by the equation $d=3\sin \frac{2\pi }{365}t+12$, where t is the number of days after March 21.

Calculation:

  $\begin{array}{c}d=3\sin \frac{2\pi }{365}t+12.\\ \text{Given}d=10\frac{1}{2}=\frac{21}{2}.\\ \frac{21}{2}=3\sin \frac{2\pi }{365}t+12\\ 3\sin \frac{2\pi }{365}t=\frac{21}{2}-12.\\ 3\sin \frac{2\pi }{365}t=-\frac{3}{2}\\ \sin \frac{2\pi }{365}t=-\frac{1}{2}.\\ \frac{2\pi }{365}t=\frac{7\pi }{6}\Rightarrow 12t=365\times 7\Rightarrow t=\frac{365\times 7}{12}\approx 213.\\ \frac{2\pi }{365}t=\frac{11\pi }{6}\Rightarrow 12t=365\times 11\Rightarrow t=\frac{365\times 11}{12}\approx 335.\end{array}$

March 21^st+213 days= October 20^th

March 21^st+335 days= February 19^th

The dates when there will be $10\frac{1}{2}$

hours of daylight will be: October 20^th, February 19^th.

b.

To determine

To find: what days of the year have at least $10\frac{1}{2}$ hours of daylight using the result in part (a).

Answer to Problem 21CYU

  $t\le 213,t\ge 335.$

The longest day of the year occurs around June 22nd, so the days between February 19th and October 20th must increase in length until June 22nd and then decrease in length until October 20th.

Explanation of Solution

Given information: The number of hours of daylight d in Hartford, Connecticut, may be

approximated by the equation $d=3\sin \frac{2\pi }{365}t+12$, where t is the number of days after March 21.

Calculation:

Have to solve the inequality:

  $d\ge 3\sin \frac{2\pi }{365}t+12.$

From the part (a)

March 21^st+213 days= October 20^th

March 21^st+335 days= February 19^th

The dates when there will be $10\frac{1}{2}$

hours of daylight will be: October 20^th, February 19^th

This means must have:

  $t\le 213,t\ge 335.$

The longest day of the year occurs around June 22nd, so the days between February 19th and October 20th must increase in length until June 22nd and then decrease in length until October 20th.