Final answer:
To find the interest rate that would grow $112,000 to $392,000 in 14 years, one must use the compound interest formula, and after calculating, solve for the rate by taking the 14th root of 3.5 and subtracting 1.
Step-by-step explanation:
To determine at what interest rate $112,000 must be invested so that it will grow to $392,000 in 14 years, we can use the formula for compound interest:
A = P(1 + r)^n
Where:
A is the amount of money accumulated after n years, including interest.
P is the principal amount (the initial amount of money).
r is the annual interest rate (decimal).
n is the number of years the money is invested for.
In this scenario, we are given:
A = $392,000
P = $112,000
n = 14
We need to solve for r, which will be in decimal form. The equation to solve is:
392,000 = 112,000(1 + r)^14
Dividing both sides by 112,000 gives us:
(1 + r)^14 = 392,000 / 112,000
(1 + r)^14 = 3.5
To find r, we will need to take the 14th root of 3.5 and then subtract 1. After calculating, we should find the interest rate that would grow $112,000 to $392,000 in 14 years.