The number of trees in each row of a garden is equal to the total number of rows in the garden. After 111 trees have been uprooted in a storm, there remain 10914 trees in the garden. The number of rows of trees in the garden is

### Concept & Logic This problem is a two-step application of algebraic modeling and square roots. We must first account for the trees lost in the storm before calculating the grid dimensions. ### Step-by-Step Solution * Let the initial number of rows in the garden be $x$. * Since the number of trees in each row equals the number of rows, the initial number of trees per row is also $x$. * Original total number of trees = $x \cdot x = x^2$. * **Given:** $111$ trees were uprooted, leaving $10914$ trees remaining. * We can set up the equation for the original total: * Original Total = Remaining Trees + Uprooted Trees * $x^2 = 10914 + 111$ * $x^2 = 11025$ * Now, solve for $x$ by finding the square root: * $x = \sqrt{11025}$ * Any number ending in $25$ has a square root ending in $5$. * Ignore the last two digits ($25$) and look at the remaining part: $110$. * We need to find two consecutive integers whose product is $110$. Those integers are $10$ and $11$ (since $10 \times 11 = 110$). * Take the smaller integer ($10$) and append the $5$. * Thus, $x = 105$. ### Exam Strategy & Shortcut Master the shortcut for squaring numbers that end in 5. The pattern for $(n5)^2$ is to compute $n \times (n+1)$ and append $25$. When you reach $x^2 = 11025$, you can immediately see the $25$ suffix. The prefix $110$ is clearly $10 \times 11$. This instantly tells you the root is $105$. No manual calculation or division is required. ### Common Pitfall A common error is setting up the equation backwards (e.g., $x^2 - 10914 = 111$), which works mathematically, but some students mistakenly subtract $111$ from $10914$ instead of adding it, finding $x^2 = 10803$ which is not a perfect square and leads to panic. Always ensure your total represents the intact original state. ### Final Answer **Therefore, the correct answer is 105.**