6.8 Try It Problems INSTRUCTOR: Assume the population of fish grows exponentially. Upon the stock initially with 500 fish, after nine months, there are 1,000 fish. The owner will allow his friends and neighbors to fish on his pond after the fish population reaches 10,000. When will the owner's friends be allowed to fish? We need to notice a few things. One is that this is growing exponentially, which means the equation we will use is y equals y0, also said y sub 0, subscript of 0. So y equals y0 e to the kt. That's the model we're going to use. Now, the pond is stocked initially with 500 fish. That means that our y0 is equal to 500. And then after nine months, there are 1,000 fish. That means we have an ordered pair that is t, comma, y. And that is equal to 9, comma, 1,000. Now, we're going to need to find the growth constant here-- solve for k, that is. So let's begin by rewriting our equation as y equals 500 e to the kt. Now we can use the values we have for t and y, 9 and 1,000 respectively, to write this equation as 1,000 equals 500 e to the k times 9. I'm going to rewrite that as 500 to the 9k. Now we wish to solve for k. We'll divide by 500 on both sides so that we have 2 equals e to the 9k. Now we can take the natural log of both sides. The natural log of e to the 9k equals natural log of 2, which means that natural log of 2 equals 9k. We'll divide by 9 on both sides so that k is equal to 1/9 natural log of 2. Now we can go back and substitute that into our original model so that our equation, putting these together, is y equals 500e to the 1/9 natural log of 2t. That is all the exponent on e-- y equals 500 e to the exponent of 1/9 natural log of 2t. Now, from this, we want to know when the population will reach 10,000. So let's replace y with 10,000. 10,000 equals 500e to the 1/9 natural log of 2t. Now we want to solve for t because we want to know when this will occur. We'll divide by 500 on both sides here so that 20 equals e to the 1/9 natural log of 2 times t. We'll take the natural log of both sides, which will result in the natural log of 20 equals 1/9 natural log of 2 times t. Solving for t, we'll divide by 1/9 natural log of 2. So this becomes t is equal to natural log of 20. So you have denominator of natural log of 2 and a numerator of 9, so t is equal to 9 natural log of 20 all over natural log of 2. Now, this is approximately 38.9. And based on our context, this is 38.9 months. [ENDS AT 4:39]