Q. 112 - Precalculus Enhanced with Graphing Utilities

Question

Difference in Time of Sunrise Naples, Florida, is approximately 90 miles due west of Ft. Lauderdale. How much sooner would a person in Ft. Lauderdale first see the rising Sun than a person in Naples? See the hint. [Hint: Consult the figure. When a person at Q sees the first rays of the Sun, a person at P is still in the dark. The person at P sees the first rays after Earth has rotated so that P is at the location Q. Now use the fact that at the latitude of Ft. Lauderdale in 24 hours an arc of length $2\mathrm{\pi }\left(3559\right)$ miles is subtended.]

Step-by-Step Solution

Answer

The time for which a person would see the Sun in St. Lauderdale than a person in Naples is 5.796 min.

Step 1. Given Information

Naples, Florida, is approximately 90 miles due west of Ft. Lauderdale.

Step 2. Formulating in mathematical terms

To solve this problem with the given 90 mile gap of the two places, the formula for linear speed is needed since the 90 mile can be considered as an arc of a circle,$v=\frac{s}{t}$ , however, the value of $v$ is unknown. Another formula of linear speed is where.

Another formula of linear speed is $v=rw where w=\frac{\theta }{t}$

Step 3. Making formula for time t .

$v=\frac{s}{t}$

$t=\frac{s}{v};v=rw$

So, $t=\frac{s}{rw}$

Step 4. Solving for time t

$t=\frac{s}{rw}$

$$\begin{gathered} t=\frac{90mi}{3559mi\left({\displaystyle \frac{1 rev}{24 hr}}\right)} \\ t=\frac{90 mi}{3559 mi \left({\displaystyle \frac{1 rev}{24 hr}}\right)\left({\displaystyle \frac{2\mathrm{\pi } \mathrm{rad}}{1 rev}}\right)} \\ t=\frac{90 mi\left(24 hr\right)}{3559 mi\left(2\mathrm{\pi } \mathrm{rad}\right)} \\ t=0.09659hr.\frac{60 min}{1 hr} \\ t=5.796 min \end{gathered}$$