I have tried hard to solve, but I'm stuck....
Let’s use our knowledge of quadrantic equations to analyze a real world business application. If P dollars is invested at r rate of interest compounded anually t years, then the amount of money, A, accummulated at the end of t years is given by the formula
A = P (1 + r)t
Barbara invested $100 at a certain rate of interest compounded anually for two years. If the accumulated value at the end of two years is $121, find the rate of interest.
To find the rate of interest, we need to rearrange the formula A = P (1 + r)^t to solve for r.
Given:
P = $100 (initial investment)
A = $121 (accumulated value at the end of two years)
t = 2 (number of years)
Substituting the values into the formula, we have:
$121 = $100 (1 + r)^2
To solve for r, we need to isolate it on one side of the equation. Let's go through the steps:
Step 1: Expand the exponent on the right side of the equation:
$121 = $100 (1 + r)(1 + r)
Step 2: Simplify the expression on the right side by multiplying:
$121 = $100 (1 + 2r + r^2)
Step 3: Distribute the $100 to each term inside the parentheses:
$121 = $100 + $200r + $100r^2
Step 4: Move all terms to one side of the equation to set it equal to zero:
$100r^2 + $200r + $100 - $121 = 0
Step 5: Combine like terms and simplify:
$100r^2 + $200r - $21 = 0
Now, we have a quadratic equation in the form of ar^2 + br + c = 0. To solve for r, we can use the quadratic formula: r = (-b ± √(b^2 - 4ac)) / 2a.
a = $100, b = $200, c = -$21
Step 6: Substitute the values into the quadratic formula and solve for r:
r = (-$200 ± √($200^2 - 4($100)(-$21))) / 2($100)
Step 7: Simplify:
r = (-$200 ± √($40,000 + $8,400)) / $200
Step 8: Continue simplifying:
r = (-$200 ± √$48,400) / $200
Step 9: Take the square root of 48,400:
r = (-$200 ± $220) / $200
Step 10: Two possible solutions:
r1 = (-$200 + $220) / $200 = $20 / $200 = 0.1
r2 = (-$200 - $220) / $200 = -$420 / $200 = -2.1
However, since we're looking for a rate of interest, it cannot be negative. Therefore, the rate of interest is 0.1 or 10%.
So, the rate of interest for Barbara's investment is 10%.