11,406.14=(438.3/r)(1+r)^10 how do you solve for r?
These are very "nasty" equations, since there is no simple-level method of solving these.
I reduced it to
(1+r)^10 = 26.02359r
If you know Calculus, you can use Newton's Method,
otherwise, try something like this
It looks like your equation could have come from the amount of an annuity formula, so the r is probably an interest rate
make a chart with 3 columns
r | Left Side|Rigth Side
.05 1.62889 1.3 ---- off quite a bit
.06 1.79084 1.5614 --- let's go higher
.09 2.36736 2.345 --- hey, not bad
.095 2.4782 2.47224 -- closer !!
.096 2.50095 2.49826
.097 2.523866 2.524288 notice the Right side is now > the left side
so our r must be between .096 and .097
try .0965
Do you get the idea?
With a good calculator and some calculator skills this is not so bad
BTW, I put 26.02359 into memory of calculator
To solve for r in the equation 11,406.14 = (438.3/r)(1+r)^10, you can use algebraic manipulations to isolate r. Here's how you can do it step-by-step:
1. Start by multiplying both sides of the equation by r to eliminate the fraction on the right-hand side:
11,406.14 × r = 438.3 × (1+r)^10
This gives you: 11,406.14r = 438.3 × (1+r)^10
2. Next, divide both sides of the equation by 438.3 to isolate the expression (1+r)^10:
(11,406.14r) / 438.3 = (438.3 × (1+r)^10) / 438.3
Simplifying, you get: 26.01r = (1+r)^10
3. Now, take the 10th root of both sides to eliminate the exponent 10 on the right-hand side:
√(26.01r) = √((1+r)^10)
This gives you: √(26.01r) = 1+r
4. Next, square both sides of the equation to remove the square root:
(√(26.01r))^2 = (1+r)^2
Simplifying, you get: 26.01r = 1 + 2r + r^2
5. Rearrange the equation to bring all terms to one side to form a quadratic equation:
r^2 + (2-26.01)r + 1 = 0
6. Use the quadratic formula to solve for r. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / 2a
In our case, substituting a = 1, b = (2-26.01), and c = 1, the formula becomes:
r = (-(2-26.01) ± √((2-26.01)^2 - 4(1)(1))) / 2(1)
Simplify and calculate the values to find the solutions for r.
Note: We recommend using a calculator or mathematical software to evaluate and compute the values accurately.