### Video Transcript

While studying the fish population in a certain lake, it was found that the number of fish in the lake decreases by 20 percent every 30 years. If the present number of fish in the lake is 500000, which of the following expressions represent the estimate of the number of fish in the lake π‘ years from now? A) 500000 times 0.8 to the π‘ over 30 power. B) 500000 times 0.8 to the 30 times π‘ power. C) 500000 times 0.2 to the π‘ over 30 power. Or D) 500000 times 0.2 to the 30 times π‘ power.

The population of fish decreases by 20 percent every 30 years. But we want to know an estimate of the number of fish in the lake. If the decrease is 20 percent every 30 years, then 80 percent remain after 30 years. If we look at options C and D, they are calculating a decrease. They arenβt calculating how many fish remain. They took the 20 percent and wrote it as a decimal, 0.2. But since weβre looking to calculate the number of fish that remain, weβll need to use that 80 percent, written as a decimal 0.8. And now we have to consider what kind of exponent we need to calculate the number of fish remaining after π‘ years.

Letβs consider the case when π‘ equals 30. We would be calculating 30 years from now. In 30 years, 80 percent of the 500000 will remain. If we plug in π‘ equals 30 for option A, we have 500000 times 0.8 to the 30 over 30 power. 30 over 30 equals one. Option A would be 500000 times 0.8. If we followed the same procedure for option B, we would have 500000 times 0.8 to the 30 times 30 power. This would mean we would take 0.8 to the 900th power. This value represents 30 sets of 30 years. Option B could not work.

We would need to take 500000 times 0.8 to the π‘ over 30 power to calculate the number of fish in the lake π‘ years from now.