Question

In a large mixing bowl, sugar is poured in at a rate of 2t + 5 cubic centimeters per
second, and milk is mixed in at a rate of 2t

2 + 1 cubic centimeters per second. After
how long does the amount of milk exceed the amount of sugar in the bowl?

Answer 1

Let $S(t)=2t+5$ - amount of sugar

and $M(t)=2t^2+1$ - amout of milk

We should solve following inequality:

$M(t)>S(t)$

$2t^2+1>2t+5 \\ 2t^2-2t-4>0\\ t^2-t-2>0$

$D=(-1)^2-4(-2)=9$

$t_1=\frac{1+3}{2}=2\\ t_2=\frac{1-3}{2}=-1$

$t\isin(-\infin,-1)\bigcup(2,+\infin)$

Since $t$ is time, it can't be less than 0.

Thereby, amount of time will exceed amount of sugar when $t>2$

Here blue line shows amount of sugar and red line shows amount of milk.