To solve this problem, let's start by assigning variables to the different unknowns. Let's call the amount of money invested in the business "x", and let's call the amount earned in the first year "y".
According to the problem, the investment earns 11/2 times as much as in the preceding year. So, in the second year, the amount earned is (11/2) * y, in the third year, it is (11/2) * (11/2) * y, and in the fourth year, it is (11/2) * (11/2) * (11/2) * y.
Now, we can set up an equation using the information given in the problem. The total amount earned in four years is $29,250. So, the equation is:
y + (11/2) * y + (11/2) * (11/2) * y + (11/2) * (11/2) * (11/2) * y = 29,250
To simplify this, we can write it as:
y + (11/2)y + (121/4)y + (1331/8)y = 29,250
To add the fractions on the left side, we need a common denominator. The common denominator is 8. So, we rewrite the equation as:
(8/8)y + (44/8)y + (121/4)y + (1331/8)y = 29,250
Now, we can combine the terms:
(1504/8)y = 29,250
To isolate "y", we can multiply both sides of the equation by 8/1504:
y = 29,250 * (8/1504)
Calculating this, we find that y ≈ $155.23 (rounded to the nearest cent).
Therefore, the amount earned in the fourth year is (11/2) * (11/2) * (11/2) * y:
(11/2) * (11/2) * (11/2) * y ≈ (11/2) * (11/2) * (11/2) * 155.23 ≈ $2,211.24 (rounded to the nearest cent).
So, the investment earned approximately $2,211.24 in the fourth year.