A student was trying to write an exponential function to represent a fish population in a local stream decreasing at a rate of 4% per year. The original population was 48,000. The student made an error when writing their exponential function.
To write the exponential function for the fish population decreasing at a rate of 4% per year, we can consider the annual decay as a decrease of 96% (100% - 4%) from the previous year's population.
The erroneous exponential function written by the student is:
P(t) = 48,000 * (0.96)^t
In this function, 'P(t)' represents the fish population after 't' years.
However, the student made an error in the exponent. Instead of writing 't' as the number of years, they mistakenly wrote it as a decimal fraction representing the decrease rate of 4%.
To correct this, we need to replace 't' with the number of years the fish population has been decreasing. Let's assume 't' stands for the number of years.
The corrected exponential function for the fish population is:
P(t) = 48,000 * (0.96)^(−0.04t)
In this function, −0.04t represents the decreasing effect of 4% per year. The negative sign ensures a decrease instead of growth, and multiplying by 0.04 gives the 4% decay rate.