An interest rate of 10% compounded continuously is desired of an investment of P15,000. How many years will be required to recover the capital with the desired interest if P2,500 is received each year?

To find out how many years will be required to recover the capital with the desired interest, we can use the formula for compound interest:

A = P * e^(rt)

Where: A = the final amount (capital + interest) P = the initial investment e = the mathematical constant approximately equal to 2.71828 r = the interest rate (in decimal form) t = the time period (in years)

In this case, we have: P = P15,000 r = 10% = 0.10 A = P15,000 + P2,500 = P17,500

Substituting these values into the formula, we get: P17,500 = P15,000 * e^(0.10t)

To solve for t, we need to isolate it. Divide both sides of the equation by P15,000: 17,500 / 15,000 = e^(0.10t)

Now, take the natural logarithm (ln) of both sides to remove the exponential: ln(17,500 / 15,000) = ln(e^(0.10t))

Using the property of logarithms, we can bring down the exponent: ln(17,500 / 15,000) = 0.10t * ln(e)

Since ln(e) is equal to 1, the equation simplifies to: ln(17,500 / 15,000) = 0.10t

Now, divide both sides by 0.10: ln(17,500 / 15,000) / 0.10 = t

Using a calculator, we can find the value of ln(17,500 / 15,000) / 0.10, which is approximately 0.1823.

Therefore, it will take approximately 0.1823 years to recover the capital with the desired interest.