Two investors, Alex and Bethany, each deposited an amount of $X into different financial plans offering a compound interest rate of 40% per annum for a duration of two years. Alex's investment is compounded semi-annually, whereas Bethany's investment is compounded annually. If Alex earns $1,136 more in total interest than Bethany at the end of the two years, find the value of X.

The correct option is $10,000.

Let the initial principal amount deposited by both Alex and Bethany be $X.

1. Calculation for Alex (Compounded Semi-Annually):
The annual interest rate is 40% per annum. Since Alex's investment is compounded semi-annually, the interest rate per half-year period is:
$\frac{40\%}{2}=20\%=0.20$
Over a duration of 2 years, there are a total of 4 semi-annual compounding periods ($2\times 2=4$ periods).
The total amount accumulated by Alex at the end of two years is:
$A_{\text{Alex}}=X{(1+0.20)}^4=X{\left(1.2\right)}^4=2.0736X$
The total interest earned by Alex is:
$I_{\text{Alex}}=A_{\text{Alex}}-X=2.0736X-X=1.0736X$

2. Calculation for Bethany (Compounded Annually):
Bethany's investment is compounded annually at a rate of 40% ($0.40$) per year for 2 years (2 annual periods).
The total amount accumulated by Bethany at the end of two years is:
$A_{\text{Bethany}}=X{(1+0.40)}^2=X{\left(1.4\right)}^2=1.96X$
The total interest earned by Bethany is:
$I_{\text{Bethany}}=A_{\text{Bethany}}-X=1.96X-X=0.96X$

3. Finding the value of X:
We are given that Alex earns $1,136 more in total interest than Bethany:
$I_{\text{Alex}}-I_{\text{Bethany}}=1136$
Substituting the interest expressions into the equation:
$1.0736X-0.96X=1136$
$0.1136X=1136$
Solving for X:
$X=\frac{1136}{0.1136}=10000$

Therefore, the value of X is $10,000.