Why does Lina decide to show Doon the paper she found in Jeanne DuPrau's The City of Ember? How do the people of Ember react to the power outage?

In chapter 7 of Jeanne DuPrau's The City of Ember, after Lina salvages what is left of the important-looking piece of paper her baby sister Poppy has chewed up, Lina shows a couple of different people. First, she shows Captain Fleery, her messenger captain, who is uninterested; then she shows her friend Lizzie, who is equally uninterested. Since she knows her grandmother had been looking for something lost that had been stored when Lina's great-great-grandfather was mayor, Lina decides it may be important enough to share with the current mayor, but he never replies to her message. In the evenings, during her free time, she studies the scraps of paper, trying to figure out what the words say. When she gets as far as recognizing the word Pipeworks, Lina decides to show Doon, since Doon works in and knows all about the Pipeworks.

Based on the word instructions that is still intact in the title, Lina knows the paper contains some sort of instructions related to the mayor's office. She also recognizes the typeset of the Builders and believes the paper contains instructions for the mayor written by the Builders. After continuing to study the scrap, she concludes the paper contains the words riverbank, door, and Pipeworks. She wonders if the paper is speaking of a door that "led to the other city" and draws the following conclusion:

The message had something to do with the river, a door, and the Pipeworks. And who did she know who knew about the Pipeworks? Doon, of course. (Ch. 7)

She then decides to show Doon the piece of paper and concludes it is the right thing to do because she knows he is serious, constantly studying things, very curious, and even wants to save the city; she knows he is the sort of person who would be willing to figure out the mystery, especially if the mystery can help save the city.