Final answer:
The present value of all future profits for the given scenario, with a continuous compound interest rate of 0.02, is $17,500,000. This calculation uses the integral of the product of the profit per year and the exponential decay based on the interest rate.
Step-by-step explanation:
To calculate the present value of all future profits for a business at a continuous compound interest rate, you would use the integral of p(t)e⁻¹ᵗᵒ with respect to time t, where p(t) represents the profit per year and r is the interest rate. Given that p(t) = $350,000 and r = 0.02 (2%), we can calculate the present value of all future profits by finding the indefinite integral of $350,000 times e to the power of -0.02t.
To calculate this, we get:
∫ 350,000e⁻0.02tdt = 350,000/⁻0.02 × e⁻0.02t
This results in:
(350,000 /⁻0.02) × e⁻0.02t = -17,500,000 × e⁻0.02t
Since we want the present value for all future profits from now (t = 0) into perpetuity, we evaluate this at the limit as t approaches infinity. The exponential term e⁻0.02t approaches zero as t becomes very large, effectively neutralizing the negative in front. Therefore, the present value of all future profits is $17,500,000.