Question

A nautical mile is a unit of distance frequently used in ocean navigation. It is defined as the length of an arc s along a great circle on the earth when the subtending angle has measure 1' = "one minute" = 1/60 of one degree. Assume the radius of the earth is 3,960 miles.

Find the length of one nautical mile to the nearest 10 feet.

Answer 1

The length of an arc can be related to the radius of circle and the angle it makes at the center of the circle by following equation:

s = rФ

Radius is given to be = 3960 miles 
We are to find the arc length in feet, so we convert the miles to feet.
1 mile = 5280 feet.
So,
Radius = 3960 x 5280 feet = 20908800 feet
Angle = 1/60 degree
The angle must be in radians before, we use its value in the equation given above.

So, 1/60 degrees in radians will be:

$\frac{1}{60} * \frac{ \pi }{180} = \frac{ \pi }{10800}$

Now we can use this value of angle in above equation to find the arc length.$s = 20908800 * \frac{ \pi }{10800} = 6080 feet$

So, rounded of to nearest 10 feet, the length of one nautical mile is 6080 feet.

Answer 2

The value of one nautical mile to the nearest feet is $\boxed{6080{\text{ feet}}}$.

Further Explanation:

The arc, radius and the angle are related to each other.

The angle can be obtained as the ratio of an arc of the circle to the radius of the circle.

The formula for the angle can be expressed as,

$\boxed{{\text{Angle = }}\frac{{{\text{length of an arc}}}}{{{\text{radius of circle}}}}}$

The length of an arc can be obtained as,

$\boxed{{\text{Length of an arc}}=\left({{\text{radius}}\times{\text{angle}}}\right)}$

Given:

The angle is one minute represented by 1'.

One minute is equal to ${\left({\frac{1}{{60}}}\right)^\circ}$.

The radius of earth is 3960 miles.

Explanation:

One mile is equal to 5280 feet.

$\boxed{1{\text{ mile}}=5280{\text{ feet}}}$.

The radius in feet can be expressed as follows,

$\begin{aligned}Radius&=3960\times5820\\&=20908800{\text{ feet}}\\\end{aligned}$

Convert the angle into radian.

$\boxed{1{\text{ degree}} = \frac{\pi }{{180}}{\text{ radian}}}$

The value of $\frac{1}{{{{60}^ \circ }}}$ can be obtained as follows,

$\begin{aligned}{\text{Angle}}&=\frac{1}{{60}}\times\frac{\pi }{{180}}\\&=\frac{\pi }{{10800}}\\\end{aligned}$

The length of an arc of the earth can be obtained as follows,

$\begin{aligned}{\text{length of an arc}}&=20908800\times\frac{\pi }{{10800}}\\&=6080{\text{ feet}}\\\end{aligned}$

Learn more:

1. Learn more about inverse of the function https://brainly.com/question/1632445.

2. Learn more about equation of circle brainly.com/question/1506955.

3. Learn more about range and domain of the function https://brainly.com/question/3412497

Answer details:

Grade: High School

Subject: Mathematics

Chapter: Unit and Measurement

Keywords: Nautical mile, unit distance, earth, radius of earth, great circle, one degree, length of nautical unit, feet.