Is $\sqrt{115}$ Between 10 And 11?

Hey guys! Today, we're diving into a cool math problem that's all about figuring out where a number sits on the number line, specifically when it's hiding under a square root sign. We're looking at the number 115 and trying to determine which of the given inequalities is true. This means we need to find out between which two consecutive whole numbers the square root of 115 lies. Let's break it down, shall we?

So, the question is: Which of the following inequalities is true? We've got four options:

A. $8< \sqrt{115}<9$ B. $9< \sqrt{115}<10$ C. $10< \sqrt{115}<11$ D. $11< \sqrt{115}<12$

To solve this, we don't actually need a calculator (though it can be a handy check!). The trick is to think about the squares of the whole numbers around our options. Remember, the square root of a number is the value that, when multiplied by itself, gives you the original number. So, if we want to know if $\sqrt{115}$ is between, say, 10 and 11, we need to check if 115 is between $10^2$ and $11^2$. Let's get our squares ready!

First up, let's test option A. Is $\sqrt{115}$ between 8 and 9? This would mean that 115 should be between $8^2$ and $9^2$. What are $8^2$ and $9^2$? Well, $8 \times 8 = 64$, and $9 \times 9 = 81$. Is 115 between 64 and 81? Nope, definitely not. 115 is way bigger than 81. So, option A is out. We can scratch that one off the list. It’s good practice to systematically eliminate options, you know?

Now, let's move on to option B. Is $\sqrt{115}$ between 9 and 10? This implies that 115 should be between $9^2$ and $10^2$. We already know $9^2 = 81$. And what's $10^2$? That's a classic: $10 \times 10 = 100$. So, is 115 between 81 and 100? Still no. 115 is larger than 100. So, option B is also incorrect. We're getting warmer, but not quite there yet, guys.

Alright, let's check out option C. This one suggests that $\sqrt{115}$ is between 10 and 11. To verify this, we need to see if 115 falls between $10^2$ and $11^2$. We just calculated $10^2 = 100$. Now, let's find $11^2$. $11 \times 11 = 121$. So, is 115 between 100 and 121? YES! 115 is indeed greater than 100 and less than 121. This means that $\sqrt{115}$ must be between $\sqrt{100}$ (which is 10) and $\sqrt{121}$ (which is 11). So, the inequality $10 < \sqrt{115} < 11$ is true! We've found our answer, but let's quickly check option D just to be absolutely sure and to reinforce our understanding.

Finally, let's look at option D. Is $\sqrt{115}$ between 11 and 12? This would mean 115 is between $11^2$ and $12^2$. We know $11^2 = 121$. What's $12^2$? $12 \times 12 = 144$. Is 115 between 121 and 144? No, it's not. 115 is less than 121. So, option D is also incorrect.

Therefore, the only inequality that holds true is option C: $10 < \sqrt{115} < 11$. It’s super satisfying when you nail these, right? By squaring the whole numbers, we were able to pinpoint the location of $\sqrt{115}$ without needing any fancy calculators. This method of using perfect squares is a fundamental skill in understanding and estimating square roots. Keep practicing, and you'll become a square root ninja in no time!

Why This Matters

Understanding inequalities and square roots isn't just about passing math tests, guys. It’s about developing your logical reasoning and problem-solving skills. When you can break down a problem like this, identify the key information (the number 115 and the inequalities), and apply a systematic approach (squaring numbers), you're building a mental toolkit that's useful in so many areas of life. Think about it – estimating costs, planning projects, even just making smart decisions relies on a similar kind of analytical thinking. Plus, mastering these concepts makes tackling more complex math problems, like those involving algebra or geometry, feel way less intimidating. It’s like building a solid foundation; the stronger it is, the higher you can build.

We looked at the number 115, but this technique works for any number. If you were asked to place $\sqrt{50}$, you’d look for squares around it. $7^2 = 49$ and $8^2 = 64$. Since 50 is between 49 and 64, $\sqrt{50}$ must be between 7 and 8. See? It’s the same principle. This estimation skill is incredibly valuable. In the real world, you often don’t need an exact decimal answer; you just need to know if something is generally large or small, or if it falls within a certain range. This ability to estimate is a superpower that engineers, scientists, financial analysts, and even everyday folks use constantly. It saves time, helps in quick decision-making, and provides a sanity check for more precise calculations.

Breaking Down the Math

Let's revisit the core idea. We're dealing with the concept of perfect squares. A perfect square is any integer that can be obtained by squaring another integer. Examples include 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on. These numbers are special because their square roots are whole numbers. When we have a number like 115 that isn't a perfect square, its square root will be an irrational number – a decimal that goes on forever without repeating. Our goal with inequalities like these is to find the two consecutive integers that sandwich this irrational number.

To do this, we list out consecutive perfect squares until we find two that bracket our target number (115 in this case). We start with smaller squares and work our way up:

• $1^2 = 1$
• $2^2 = 4$
• ... (skipping a bunch)
• $8^2 = 64$
• $9^2 = 81$
• $10^2 = 100$
• $11^2 = 121$
• $12^2 = 144$

Looking at this list, we see that 115 falls between 100 and 121. This is the crucial step. Because 115 is between 100 and 121, its square root, $\sqrt{115}$, must be between the square roots of 100 and 121.

Mathematically, if $a < b < c$, then $\sqrt{a} < \sqrt{b} < \sqrt{c}$ (assuming a, b, and c are positive). Applying this to our situation:

Since $100 < 115 < 121$, Then $\sqrt{100} < \sqrt{115} < \sqrt{121}$.

We know that $\sqrt{100} = 10$ and $\sqrt{121} = 11$.

Therefore, $10 < \sqrt{115} < 11$.

This confirms that option C is the correct answer. This methodical approach, relying on the properties of square roots and inequalities, is fundamental to understanding numerical relationships in mathematics. It’s not just about getting the right answer; it’s about understanding why it's the right answer, which is the real key to mathematical mastery. So, next time you see a square root, think about its neighboring perfect squares – it's your shortcut to understanding its value!