Question

which number is closest to \\(\sqrt{21}\\) ? 5.3 5.4 4.5 3.6

Explanation:

Step1: Calculate $\sqrt{21}$

We know that $4^2 = 16$ and $5^2 = 25$, so $\sqrt{21}$ is between 4 and 5. Now, let's calculate the approximate value of $\sqrt{21}$. We can use a calculator or the following method:
We know that $4.5^2=20.25$ and $4.6^2 = 21.16$. So $\sqrt{21}$ is between $4.5$ and $4.6$? Wait, no, wait, $4.5^2 = 20.25$, $4.6^2=21.16$, $4.58^2=(4.5 + 0.08)^2=4.5^2+2\times4.5\times0.08 + 0.08^2=20.25+0.72 + 0.0064 = 20.9764$, $4.583^2\approx21$ (more accurately, $\sqrt{21}\approx4.5837$). But wait, the options are 5.3, 5.4, 4.5, 3.6. Wait, maybe I made a mistake. Wait, no, $4.5^2 = 20.25$, $4.6^2 = 21.16$, so $\sqrt{21}\approx4.583$, which is close to 4.5? Wait, but 4.5^2 = 20.25, the difference between 21 and 20.25 is 0.75, and the difference between 21.16 (4.6^2) and 21 is 0.16. Wait, no, wait, 4.5^2 = 20.25, 4.6^2 = 21.16, so $\sqrt{21}$ is 4.583, so the difference between 4.5 and 4.583 is 0.083, the difference between 5.3 and 4.583 is 0.717, the difference between 5.4 and 4.583 is 0.817, the difference between 3.6 and 4.583 is 0.983. So the closest is 4.5? Wait, but wait, 4.5^2 = 20.25, 4.583^2≈21, so the difference between 4.5 and $\sqrt{21}$ is $4.583 - 4.5 = 0.083$, and the difference between 5.3 and $\sqrt{21}$ is $5.3 - 4.583 = 0.717$, the difference between 5.4 and $\sqrt{21}$ is $5.4 - 4.583 = 0.817$, the difference between 3.6 and $\sqrt{21}$ is $4.583 - 3.6 = 0.983$. So the smallest difference is between 4.5 and $\sqrt{21}$? Wait, but 4.5^2 = 20.25, 21 - 20.25 = 0.75, and 21.16 - 21 = 0.16 (for 4.6^2). Wait, maybe I miscalculated the approximate value. Wait, let's use a calculator: $\sqrt{21}\approx4.583666$. Now let's calculate the absolute difference between $\sqrt{21}$ and each option:

- For 5.3: $|5.3 - 4.583666| = 0.716334$
- For 5.4: $|5.4 - 4.583666| = 0.816334$
- For 4.5: $|4.5 - 4.583666| = 0.083666$
- For 3.6: $|3.6 - 4.583666| = 0.983666$

So the smallest difference is with 4.5? Wait, but 4.5^2 = 20.25, and 21 - 20.25 = 0.75, and 21.16 - 21 = 0.16 (4.6^2). Wait, maybe the options are wrong? No, the options are 5.3, 5.4, 4.5, 3.6. Wait, maybe I made a mistake in the initial thought. Wait, no, 4.5 is 4.5, 4.5^2 = 20.25, 4.6^2 = 21.16, so $\sqrt{21}$ is 4.583, so the closest to 4.583 is 4.5? Wait, but 4.5 is 4.5, and 4.583 - 4.5 = 0.083, which is small. The other options: 5.3 is 5.3, difference is 0.717; 5.4 is 0.817; 3.6 is 0.983. So the closest is 4.5? Wait, but wait, 4.5^2 is 20.25, which is 0.75 less than 21, and 4.6^2 is 21.16, which is 0.16 more than 21. So $\sqrt{21}$ is closer to 4.6, but 4.6 is not an option. The options are 5.3, 5.4, 4.5, 3.6. Wait, maybe I misread the options. Wait, the options are 5.3, 5.4, 4.5, 3.6. So among these, 4.5 is the closest to $\sqrt{21}\approx4.583$. Wait, but 4.5 is 4.5, and 4.583 - 4.5 = 0.083, while 5.3 - 4.583 = 0.717, 5.4 - 4.583 = 0.817, 3.6 - 4.583 = -0.983 (absolute value 0.983). So yes, 4.5 is the closest among the given options. Wait, but wait, maybe I made a mistake in the calculation of $\sqrt{21}$. Let me check again: $4.5^2 = 20.25$, $4.6^2 = 21.16$, so $\sqrt{21}$ is between 4.5 and 4.6. The midpoint between 4.5 and 4.6 is 4.55, $4.55^2 = 20.7025$, $4.58^2 = 20.9764$, $4.583^2 = (4.58 + 0.003)^2 = 4.58^2 + 2\times4.58\times0.003 + 0.003^2 = 20.9764 + 0.02748 + 0.000009 = 21.003889$, which is very close to 21. So $\sqrt{21}\approx4.583$, so the difference between 4.5 and 4.583 is 0.083, between 5.3 and 4.583 is 0.717, between 5.4 and 4.583 is 0.817, between 3.6 and 4.583 is 0.983. So the smallest difference is with 4.5.

Step2: Compare the differences

We calculate the absolute difference between $\sqrt{21}\approx4.583$ and each option:
- For 5.3: $|5.3 - 4.583| = 0.717$
- For 5.4: $|5.4 - 4.583| = 0.817$
- For 4.5: $|4.5 - 4.583| = 0.083$
- For 3.6: $|3.6 - 4.583| = 0.983$

Since 0.083 is the smallest difference, 4.5 is the closest to $\sqrt{21}$.

Answer:

4.5