Lily pads on the MSU lake double everyday. If the lake is completely covered on day 30, on what day was the pond half full?

Question

Lily pads on the Mississippi State lake double everyday.  If the lake is completely covered on day 30, on what day was the pond half full?  I don't understand.  wouldn't it be day 15.  How do I write an equation for that.  Is this right?

Answer 1

Hi Courtney,

 

This problem is tricky because it's difficult to wrap one's head around just how fast doubling can grow an amount of something, like a population of lily pads.  In this case, with the lily pads doubling every day, it doesn't really matter how big the pond is because the rate of growth is proportional to the size of the population.  This sort of relationship is often referred to as exponential growth, and we can see it at work in real life in bacteria populations, compound interest, radioactive decay, and more.  I'll put some more detailed equations below, but the question you asked is more of a logic puzzle, and we can reason it out without having to do any algebra.

 

Think about the pond as if it were a pie, with the lily pads representing thin slices of area of roughly equal size. Each day, the number of slices, and thus the total area covered by the lily pads, doubles; so the fraction of the pie covered by the lily pads will also double.  If the pond is completely full (or we have a whole pie) on day 30, the fraction must have doubled over the course of the final day, meaning the pond was half full on day 29!  This really emphasizes just how rapid exponential growth can be for large populations.

 

Let's work through the problem algebraically:

 

We want to study the area covered by the lily pads, so let's write that area as a function of time, A(t).

We know the area doubles every day, so if on day t we have some amount, on day t+1, we will have twice that

So, we write A(t+1) = 2*A(t).

 

In order to find the exact form of A(t), we'd have to use calculus, but we can work through the rest of the problem using algebra:

Let's write the area of the surface of the pond as P, and we know that on day 30 the lily pads cover that whole area.

Then we want to find the day number x where the lily pads cover half that area, which is P/2.

So we write down what we have:

 

The growth relation

A(t+1) = 2*A(t)

 

The known endpoint

A(30)=P

 

And the objective

A(x)=P/2

 

We also know we're starting from a time greater than x, so we'll plug the known point into the growth relation like this:

A(t+1) = A(30) = P = 2*A(t)

 

This lets us compare the arguments t+1=30 to get t=29 for the right hand side

So

A(30) = P = 2*A(29)

 

Now we can divide both sides by 2 to get

P/2 = A(29)

 

But that was what we were trying to solve for, our objective, A(x) = P/2.  So x=29.

 

Like I said before, the calculus needed to derive an exact formula for A(t) is beyond the scope of the problem, but for the sake of completeness, I'll mention a formula that works for the growth relation and points discussed above.

 

A(t) = P*2^t-30

 

In practice, though, the lily pads will stop dividing so quickly once the lake is full, and you can't have fewer than 1 lily pad, so this A(t) won't work for any value of t you plug in.  What happens on day 31, for instance?  A(31) = 2*A(30) = 2*P, but no real lake can hold twice its area worth of lily pads!!  So the math is only as real as the problem...

Answer 2

 

 

If it is full on the 30th day

 

then it is half full the day before.

 

I have shown you the equations in the other posting.

 

Please re-read them