c Set the ratio of the previous arc to trip length assumption equal to the ratio of the new arc to trip length. Solve for the trip length based off the calculated arc length in Part B.
A
a x^2+y^2=15 681 600
B
b 69.1 miles
C
c About 32 days
Practice makes perfect
a The standard form for the equation of a circle with center ( h, k) and radius r is (x- h)^2+(y- k)^2= r^2. We are told that the center is at the origin ( 0, 0) with radius 3960 miles. Let's substitute these values into the standard form to find the equation of the equator.
(x- h)^2+(y- k)^2= r^2
⇕
(x- 0)^2+(y- 0)^2=( 3960)^2
⇕
x^2+y^2=15 681 600
The equation of the equator is at x^2+y^2=15 681 600.
b To find the length of the arc, we will find what proportion it makes up of the total length of the equator. The total length of a circle is represented by its circumference. Recall the formula for the circumference of a circle.
C=2π r
Let's substitute the radius 3960 miles to determine the circumference of the equator.
The ratio of the length of the arc to the circumference of the equator is equivalent to the ratio of the degree of the arc to the total degrees of the equator.
Arc_L/Equator_C = Arc °/Equator °
The equator, which is represented by a circle, is equivalent to 360 degrees. Let's substitute the known values to solve for the length of the arc.
The length of a 1 ° arc on the equator is about 69.1 miles.
c Assuming the length of the arc is 45 miles long, Columbus thought the trip would take 21 days.
Old arc length assumption/Old trip length assumption=New estimated arc length/New estimated trip length
Let's substitute the known values to find a new estimate of length of the trip.
