The following are the ages (years) of 5 people in a room:
13, 22, 13, 18, 14
A person enters the room.
The mean age of the 6 people is now 23.
What is the age of the person who entered the room?

58

Step-by-step explanation:

the mean is adding up all the numbers, then dividing by the amount of the numbers.

add up all the numbers:

13 + 13 + 14 + 18 + 22 = 80

the mean is 23.

so you need to back track:

23 * 6 = 138

138 - 80 = 58

x = central angle apc = measure of minor arc ac

the two angles x and 30 add to 180, so x+30 = 180 and x = 180-30 = 150. the measure of minor arc ac is 150 degrees.

what we need to do from here is find the circumference and then multiply it by x/360 since there are 360 degrees in a circle. in this case, x = 150, so x/360 = 150/360 = 15/36 = 5/12. the shortest distance along the curve from a to c is 5/12 of the distance around the entire circle (aka circumference)

c = circumference

c = pi*d

c = pi*24

c = 24pi

now multiply this by 5/12 to get

distance along minor arc ac = (5/12)*(circumference)

distance along minor arc ac = (5/12)*(24pi)

distance along minor arc ac = 10pi

the formula you can use is

l = (x/360)*pi*d

with

l = length of minor arc

x = central angle

d = diameter

which essentially is the steps we took though a bit differently.