To solve the equation [tex]\( 15,730,400 = 7,400,000 \left(1 + \frac{R}{100} \right)^2 \)[/tex] for [tex]\( R \)[/tex], follow these steps. This will require understanding exponential growth and solving quadratic equations.
### Step-by-Step Solution:
1. Set Up the Equation:
Start with the given equation:
[tex]\[ 15,730,400 = 7,400,000 \left(1 + \frac{R}{100} \right)^2 \][/tex]
2. Isolate the Exponential Term:
Divide both sides of the equation by [tex]\( 7,400,000 \)[/tex]:
[tex]\[ \frac{15,730,400}{7,400,000} = \left(1 + \frac{R}{100} \right)^2 \][/tex]
Simplify the fraction:
[tex]\[ \frac{15,730,400 \div 7,400,000}{7,400,000 \div 7,400,000} = \left(1 + \frac{R}{100} \right)^2 \][/tex]
[tex]\[ \frac{15,730,400}{7,400,000} = \left(1 + \frac{R}{100} \right)^2 \][/tex]
[tex]\[ \frac{15730400}{7400000} \approx 2.12541 \][/tex]
Let:
[tex]\[ \ 2.12541 = \left(1 +\frac{R}{100} \right)^2 \][/tex]
3. Remove the Square by Taking the Square Root:
Take the square root of both sides:
[tex]\[ \sqrt{2.12541} = \sqrt{\left(1 + \frac{R}{100} \right)^2} \][/tex]
Don't forget the plus and minus when you take the square root.
[tex]\[ \pm \sqrt{2.12541} = 1 + \frac{R}{100} \][/tex]
4. Solve for [tex]\( R \)[/tex]:
Solve individually for the positive and negative roots.
- For the positive root:
[tex]\[ \sqrt{2.12541} \approx \frac{106 \sqrt{2590}}{37} \][/tex]
[tex]\[ \frac{R}{100} = \frac{106 \sqrt{2590}}{37}- 1 \][/tex]
[tex]\[ R = 100 \left(\frac{106 \sqrt{2590}}{37} - 1\right) \][/tex]
Simplifying further, we get:
[tex]\[ R = -100 + \frac{106 \sqrt{2590}}{37} \][/tex]
- For the negative root:
[tex]\[ -\sqrt{2.12541} = -\frac{106 \sqrt{2590}}{37} \][/tex]
[tex]\[ \frac{R}{100} = -\frac{106 \sqrt{2590}}{37} - 1 \][/tex]
[tex]\[ R = 100 \left(-\frac{106 \sqrt{2590}}{37} - 1\right) \][/tex]
Simplifying further, we get:
[tex]\[ R = -\frac{106 \sqrt{2590}}{37} - 100 \][/tex]
### Conclusion:
The two possible solutions for [tex]\( R \)[/tex] that solve the equation [tex]\( 15,730,400 = 7,400,000 \left(1 + \frac{R}{100} \right)^2 \)[/tex] are:
[tex]\[ R = -100 + \frac{106 \sqrt{2590}}{37} \][/tex]
[tex]\[ R = -\frac{106 \sqrt{2590}}{37} - 100 \][/tex]
These values represent the possible rates [tex]\( R \)[/tex] that satisfy the given financial equation over the specified period.
