Miguel needs 1/2 gallons of milk to make 12 milkshakes. How much does he need to make 42 milkshakes?

he needs 17/8 gallons of milk to make 30 milkshakes and this is proportional relationship

Step-by-step explanation:

3/4 divided by 12 = 1/16

gallon to make 1 milkshake and

1/16 x 30 = 17/8 gals. to make 30 milkshakes,

it's proportional because 3:4 > 15:8

step-by-step explanation:

part a. the differences are decreasing, so we know the sequence is not arithmetic. the ratios are constant (=3/4), so we know the sequence is geometric.

part b. from part a, we know the common ratio is 3/4, so each term is 3/4 of the one before. the initial term is the number of gallons at h=1.

g[h] = (3/4)·g[h-1]

g[1] = 384 . . . . initial value required for a recursive definition

part c. the general formula for a geometric sequence is

a[n] = a[1]·r^(n-1)

we have a[1] = 384, r = 3/4, and the term number given by h. the formula that follows is

g[h] = 384·(3/4)^(h-1)

part d. filling in the value and doing the arithmetic, we have

g[12] = 384·(3/4)^11 ≈ 384·0.042235136 ≈ 16.2183 . . gallons

part e. each gallon is 4 quarts, so

16.2183 gal × 4 qt/gal = 64.8732 qt