## Task B-2.2.

Everyone knows that Pinocchio's nose grows when he lies. The longer and faster Pinocchio lies, the more his nose grows. One day, Pinocchio Piko lied so long that his nose grew by $54 \mathrm{~mm}$. Pinocchio Niko lied long enough for his nose to grow by $6 \mathrm{~mm}$. Piko lied 4 minutes longer than Niko. If Piko had lied as long as Niko, and Niko as long as Piko, their noses would have grown by the same amount. Determine how long each of them would need to lie for their noses to grow equally and how much their noses would grow in that case.
## Solution.

Let Piko have lied for $t+4$ minutes, and Niko for $t$ minutes

Piko: $\left\{\begin{array}{ll}54 \mathrm{~mm} \ldots & t+4 \\ x \mathrm{~mm} \ldots & t\end{array} \Rightarrow x=\frac{54 t}{t+4}\right.$.

Niko: $\left\{\begin{array}{ll}6 \mathrm{~mm} \ldots & t \\ x \mathrm{~mm} \ldots & t+4\end{array} \Rightarrow x=\frac{6(t+4)}{t}\right.$.

$$
\Rightarrow \frac{54 t}{t+4}=\frac{6(t+4)}{t}
$$

1 point

$$
\Rightarrow t^{2}-t-2=0
$$

The solutions to the equation are $t_{1}=-1, t_{2}=2$ so $t=2$ minutes.

If Piko lied for 2 minutes, and Niko for 6 minutes, their noses would have grown $18 \mathrm{~mm}$.