Final answer:
The student's question asks for the future value of a sculpture that appreciates at a rate of $100e^t per year for 12 years, representing an application of continuous compound interest. The calculation involves integrating the rate of appreciation over the 12-year period and adding this to the initial value of the sculpture to find its worth after 12 years.
Step-by-step explanation:
The question concerns the calculation of the future value of an investment at a continuously compounding rate, specifically within the context of the appreciation of a sculpture. The sculpture's value increases at a rate represented by $100e^t per year, where e is the base of the natural logarithm and t is time in years.
To determine the sculpture's worth in 12 years, we apply the formula for continuous compounding, which is P = Pe^rt, where P is the initial principal amount, r is the rate of increase, and t is the time in years. However, in this case, since the rate is given in terms of e, we do not need to use r separately in our computation. Instead, we'll use the growth rate function provided of $100e^t.
We would integrate this function over the time period from 0 to 12 to find the total increase in value over the 12 years. The integration of $100e^t from 0 to 12 gives the total increase, which when added to the initial value of the sculpture, yields the final value after 12 years.
An important concept related to this type of problem is compound interest, which represents the addition of interest to the principal sum of a loan or deposit. In this context, the exponential function demonstrates compound interest at a continuous rate.