Ronelio invests $29,000 in two one-year certificates of deposit. One certificate pays 4%, and the other pays 4.75% simple interest annually.
(a) Construct a model for the total interest I(x) Ronelio earns in one year on his investments. (Let x represent the amount invested at 4%.)
(b) If Ronelio's total interest is $1332.50, how much money did he invest in each certificate?
(a)
i(x) = .04 x + .0475 (29,000 - x)
= 1377.50 - .0075 x
(b)
1332.50 = 1377.50 - .0075 x
To solve this problem, we can set up two equations based on the given information.
Let's assume Ronelio invests x dollars at 4% interest, so he invests (29000 - x) dollars at 4.75% interest.
(a) Model for total interest:
The interest earned on the amount invested at 4% is 4% of x, or 0.04x.
The interest earned on the amount invested at 4.75% is 4.75% of (29000 - x), or 0.0475(29000 - x).
Therefore, the model for the total interest earned, I(x), is the sum of these two interests:
I(x) = 0.04x + 0.0475(29000 - x)
(b) To find the amount invested in each certificate when the total interest is $1332.50, we need to solve the equation:
1332.50 = 0.04x + 0.0475(29000 - x)
To solve this equation, we can simplify and solve for x:
1332.50 = 0.04x + 1377.5 - 0.0475x
1332.50 - 1377.50 = 0.04x - 0.0475x
-45 = -0.0075x
x = -45 / -0.0075
x = 6000
Therefore, Ronelio invested $6000 at 4% interest, and the remaining amount, 29000 - 6000 = $23000, was invested at 4.75% interest.