Smalltown, latitude 53.6 degrees N, lies directly north of Bigville, latitude 38.6 degrees N. Give the distance between the two cities. (Hint: The earth's radius is about 3,960 miles.)

This is a question similar to what will be on my test tomorrow. I'd really appreciate the help on how to figure this out.

The answer is 330 Pi or 1,036.7 miles.

draw a 'great circle'

the two towns form a sector with an angle
of 15º (53.6-38.6 = 15)
15º = pi/12 radians

you want the arclength of that sector
arclength = radius x angle
= 3960(pi/12) = 330pi

ggyu

To find the distance between Smalltown and Bigville, we can use the formula for the distance between two points on a sphere, such as the Earth.

The formula for the distance between two points on a sphere is given by the Haversine formula:

d = 2r * arcsin(sqrt(sin^2((lat2 - lat1)/2) + cos(lat1) * cos(lat2) * sin^2((lon2 - lon1) / 2))), where
- d is the distance between the two points,
- r is the radius of the Earth,
- lat1 and lat2 are the latitude coordinates of the two cities, and
- lon1 and lon2 are the longitude coordinates of the two cities.

Now, let's calculate the distance between Smalltown (latitude 53.6 degrees N) and Bigville (latitude 38.6 degrees N) using the given radius of the Earth (approximately 3,960 miles).

Step 1: Convert the latitude coordinates from degrees to radians:
- Smalltown (lat1) = 53.6 degrees * (pi/180) = 0.935 radians
- Bigville (lat2) = 38.6 degrees * (pi/180) ≈ 0.674 radians

Step 2: Substitute the values into the Haversine formula:
d = 2 * 3960 * arcsin(sqrt(sin^2((0.674 - 0.935)/2) + cos(0.935) * cos(0.674) * sin^2(0)))

Step 3: Simplify the formula:
d = 2 * 3960 * arcsin(sqrt(sin^2(-0.130)/2 + cos(0.935) * cos(0.674) * sin^2(0)))

Step 4: Evaluate the trigonometric functions:
d = 2 * 3960 * arcsin(sqrt(0.015161 + cos(0.935) * cos(0.674) * 0))

Step 5: Calculate the trigonometric functions:
d = 2 * 3960 * arcsin(sqrt(0.015161 + 0.585152 * 0))
d = 2 * 3960 * arcsin(sqrt(0.015161))

Step 6: Simplify the expression within the square root:
d = 2 * 3960 * arcsin(0.123283)

Step 7: Calculate the arcsin (inverse sine) and convert from radians to degrees:
d = 2 * 3960 * 7.0479 ≈ 27,835.68 miles

So, the distance between Smalltown and Bigville is approximately 27,835.68 miles.

The answer you provided, 1,036.7 miles or 330π, is not correct given the latitude coordinates and the Earth's radius provided in the question. Please double-check the values and calculations.