Which of the following equations can be used to model the number of meals someone will eat over a period of time if they eat 3 meals a day.
3t + 5 = 32
4- 3t = 6
2 + 6t = 8
3t+ 2 = 20
2t + 2 = 6

3t + 5 = 32 and 3t+ 2 = 20

Step-by-step explanation:

We know that someone eats 3 meals per day.

Then if we define the variable t, as the number of days passed since we started counting, we can write the total number of meals eaten as:

M(t) = 3*t

With that in mind, we only can have multiples of 3 (and the coefficient must be 3)

Then the options that remain are:

1) 3*t + 5 = 32

2) 4 - 3*t = 6

3) 3*t + 2 = 20.

Now, let's go to our linear equation:

M(t) = 3*t

As t can only take whole numbers, we can see that M(t) can only be a multiple of 3.

Then we must have:

3*t = multiple of 3.

So let's see our options:

1) 3*t + 5 = 32

3*t = 32 - 5 = 27

27 is a multiple of 3, such that 3*9 = 27, this means that in 9 days, this person would eat 27 meals, then this equation can represent our situation.

2) 4 - 3*t = 6

 3*t = 4 - 6 = -2

3*t = -2

-2 is not a multiple of 3 and is a negative number, so this does not make any sense with our initial situation.

3) 3*t + 2 = 20

3*t = 20 - 2 = 18

3*t = 18

18 is a multiple of 3, such that 3*6 = 18

This says that in 6 days, this person would eat 18 meals, then this equation can represent the situation of this problem.

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