A lily pad doubles in area every second. After one minute, it fills the pond. How long would it take to quarter fill the pond ? A lily pad doubles in area every second. After one minute, it fills the pond. How long would it take to quarter fill the pond?


To me this seems like we can set up a fraction-like equation:

$$\frac{60 \ \text{seconds}}{1} = \frac{x \ \text{seconds}}{1/4}$$ then $x = 15$ seconds. But the answer is $58$ seconds which really makes no sense to me. Any suggestions are greatly appreciated.
I think it's easiest to work backwards: if the area doubles every second and the pond is totally covered at time $t=60$, then it must be half covered at $t=59$, and therefore one quarter covered at $t=58$.

Alternately, let $f(t)$ be the fraction of the pond's area covered at time $t\leq 60$. Then $f(t)=f(0)2^t$ since the area doubles every second, and since $f(60)=1$ we get $f(0)=2^{-60}$. Therefore $f(t)=2^{-60}2^t=2^{t-60}$. Then setting $ 2^{t-60}=\frac{1}{4}$ and solving for $t$ yields $t=58$.
The area of a lily pod can be described by a function:

$$f:[0,60]\to[0,1]\\
f(x)=2^{x-60}$$

Now we have to compute the argument $x$ for which the value of the function $f(x)$ is equal to $\frac{1}{4}$:

$$\frac{1}{4}=2^{x-60}\\
\log_2 \frac{1}{4}=\log_2 2^{x-60}\\
-2 = x-60\\
x=58$$