The number of marine organisms in a pool, 𝑦, after 𝑛 weeks is given by the formula 𝑦 equals 4344 multiplied by a half to the power 𝑛 minus one. After how many weeks are there 1086 marine organisms in the pool?
In order to answer the question and solve the equation to find 𝑛, we’re first gonna have a look at what information we’ve got. Well, first of all, we know, looking back at the question, that the number of marine organisms in a pool is donated by 𝑦. And then we’re told later on that actually there are 1086 marine organisms in the pool. So therefore, we can say that 𝑦 is equal to 1086. Then we move on to 𝑛. And we can see that 𝑛 is actually the unknown we want to find because we know that 𝑛 is the weeks. And actually, we want to know after how many weeks are there 1086 marine organisms in the pool.
Okay, great. So now we have those, and we know what we’re looking for. We can substitute it back into the equation and start to solve. So first of all, we get 1086 is equal to 4344 multiplied by a half to the power of 𝑛 minus one. So then, what we do is we actually divide both sides by 4344, which gives us 1086 over 4344 is equal to a half to the power of 𝑛 minus one. But then we can actually simplify because 1086 divided by 4344 is actually a quarter. So, great. So we now have that a quarter is equal to a half to the power of 𝑛 minus one.
And now, the next stage that we’re going to, in order to solve this, is actually what we’re gonna do is we’re gonna make the bases the same. So I’m gonna rewrite a quarter in terms of a half. So what I have actually done is I’ve rewritten a quarter as a half squared because a half multiplied by a half gives us a quarter. So now you can actually see that we actually have both sides of the equation with the same base, which is a half. But now, we’ve got a half to the power of two, or a half squared, is equal to a half to the power of 𝑛 minus one.
And now, as we have the same bases, what we can actually do is we can actually equate the exponents cause we can say that two is equal to 𝑛 minus one. And like I said, that’s because we had a half to the power of two and a half to the power of 𝑛 minus one.
Okay, great. So we can now do this and solve to find 𝑛. And therefore, if we add one to each side, we’re to get 𝑛 is equal to three. So therefore, we can say that after three weeks, there are gonna be 1086 marine organisms in the pool.