Mr. Hammond is the volunteer coordinator for a company that puts on running races. Last year, the company organized 7 short races and 8 long races, which required a total of 804 volunteers. This year, the company organized 11 short races and 5 long races, which required 688 volunteers in total. How many volunteers does each type of race require?

28 short race volunteers

76 long race volunteers

Step-by-step explanation:

Let :

Short races = a

Long races = b

LAST YEAR:

7a + 8b = 804 - - - - (1)

THIS YEAR:

11a + 5b = 688 - - - (2)

Multiply (1) by 5 and (2) by 8

35a + 40b = 4020

88a + 40b = 5504

Subtract

-53a = - 1484

a = 1484 / 53

a = 28

Put a = 28 in equation (1)

7(28) + 8b = 804

196 + 8b = 804

8b = 804 - 196

8b = 608

b = 76

28 short race volunteers

76 long race volunteers

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.

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step-by-step explanation: