A vending machine coin box contains $22.80 in nickels,

24 nickels, 36 dimes, and 72 quarters

Pembahasan

Let n be the number of nickels, d be the number of dimes, and q be the number of quarters.

From the problem statement, we have the following equations:
1. Total number of coins: n + d + q = 132
2. Total value of coins: 0.05n + 0.10d + 0.25q = 22.80
3. Relationship between quarters and dimes: q = 2d

Now we can solve this system of equations.
Substitute equation (3) into equation (1):
n + d + (2d) = 132
n + 3d = 132
Solve for n: n = 132 - 3d

Substitute equation (3) into equation (2):
0.05n + 0.10d + 0.25(2d) = 22.80
0.05n + 0.10d + 0.50d = 22.80
0.05n + 0.60d = 22.80

Now substitute the expression for n (from n = 132 - 3d) into this equation:
0.05(132 - 3d) + 0.60d = 22.80
6.6 - 0.15d + 0.60d = 22.80
6.6 + 0.45d = 22.80
0.45d = 22.80 - 6.6
0.45d = 16.2
d = 16.2 / 0.45
d = 36

Now that we have the number of dimes, we can find the number of quarters using equation (3):
q = 2d = 2 * 36 = 72

Finally, we can find the number of nickels using equation (1):
n + d + q = 132
n + 36 + 72 = 132
n + 108 = 132
n = 132 - 108
n = 24

So, the box contains 24 nickels, 36 dimes, and 72 quarters.