How much grape juice would you need to mix with 15/4 quarts of sparkling water. (a recipe for sparkling grape juice calls for 1 1/2 quarts of sparkling water and 3/4 quart of grape juice)

15/8 quarts

Step-by-step explanation:

The recipe calls for 1 1/2 quarts of sparkling water.  This can be written as the improper fraction 3/2 (this is because 1 whole = 2/2; 2/2 + 1/2 = 3/2).

Using 15/4 quarts sparkling water means we used 5/2 times as much as the recipe calls for.  This means we would use 5/2 times the amount of grape juice as well; 5/2(3/4) = 15/8.

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.