Chapter 7.1, Problem 75AYU

Being the First to See the Rising Sun Cadillac Mountain, elevation 1530 feet, is located in Acadia National Park. Maine, and is the highest peak on the east coast of the United States. It is said that a person standing on the summit will be the first person in the United States to see the rays of the rising Sun. How much sooner would a person atop Cadillac Mountain see the first rays than a person standing below, at sea level?

[ Hint: Consult the figure. When the person at $D$ sees the first rays of the Sun, the person at $F$ does not. The person at $F$ sees the first rays of the Sun only after Earth has rotated so that $F$ is at location $Q$. Compute the length of the arc subtended by the central angle $\theta$. Then use the fact that at the latitude of Cadillac Mountain, in 24 hours a length of $2\pi \left(2710\right)\approx 17,027.4$ miles is subtended.]

To determine

To find: How much sooner would a person atop Cadillac Mountain see the first rays than a person standing below, at sea level.

Answer to Problem 75AYU

$3.35$ mins

Explanation of Solution

Given:

Cadillac Mountain, elevation 1530 feet, is located in Acadia National Park, Maine, and is the highest peak on the east coast of the United States. It is said that a person standing on the summit will be the first person in the United States to see the rays of the rising Sun.

Calculation:

Radius of the earth is 14308800 feet.

Length of $AD$ is 14310330 feet.

The arc angle is ${\cos }^{-1}\left(\frac{14308800}{14310330}\right)$.

$\text{DAQ }=0.8378\text{ degrees }=\text{ 0}.0146\text{ rad }$

$\text{Arc length }\left(S\right)=r\theta =0.0121\times 2710=39.63\text{ miles }$

Circumference of the earth $=2\pi r=2\pi \times 2710=17027$ miles

How much sooner would a person atop Cadillac Mountain see the first rays than a person standing below, at sea level,

$\frac{24\times 60}{17027}=\frac{x}{39.63}$

$x=3.35$ mins