Question

which number is closest to $-sqrt{148}$? -11.5 -12.1 -13.4 -13.3

Explanation:

Step1: Estimate $\sqrt{148}$

We know that $12^2 = 144$ and $13^2 = 169$. Since $144<148<169$, $\sqrt{144}<\sqrt{148}<\sqrt{169}$, so $12<\sqrt{148}<13$. Now, calculate the difference between $148$ and $144$: $148 - 144 = 4$. The difference between $169$ and $144$ is $169 - 144 = 25$. So $\sqrt{148}\approx12+\frac{4}{25}\times1 = 12 + 0.16 = 12.16$? Wait, no, that's a wrong way. Actually, we can calculate $12.1^2=146.41$, $12.2^2 = 148.84$. Since $12.1^2 = 146.41$ and $12.2^2=148.84$, and $148 - 146.41 = 1.59$, $148.84 - 148 = 0.84$. So $\sqrt{148}$ is closer to $12.2$? Wait, no, wait $12.1^2 = 146.41$, $12.1^2=146.41$, $12.1\times12.1 = (12 + 0.1)^2=12^2 + 2\times12\times0.1+0.1^2 = 144 + 2.4 + 0.01 = 146.41$. Then $12.2^2=(12 + 0.2)^2=144 + 4.8 + 0.04 = 148.84$. So $148$ is between $12.1^2$ and $12.2^2$. The distance from $148$ to $146.41$ is $148 - 146.41 = 1.59$. The distance from $148.84$ to $148$ is $148.84 - 148 = 0.84$. So $\sqrt{148}$ is closer to $12.2$? Wait, but the options are -11.5, -12.1, -13.4, -13.3. Wait, maybe I made a mistake. Wait, $13^2 = 169$, $12^2=144$, $13^2 - 148 = 169 - 148 = 21$, $148 - 12^2 = 4$. Wait, no, $12^2=144$, $13^2=169$, so $\sqrt{148}$ is between 12 and 13. Wait, but the options have -13.3 and -13.4, which are around -13. So maybe I miscalculated. Wait, wait, $12.1^2 = 146.41$, $12.2^2=148.84$, so $148$ is between $12.1^2$ and $12.2^2$. So $\sqrt{148}\approx12.17$ (since $148 - 146.41 = 1.59$, $148.84 - 146.41 = 2.43$, so $1.59/2.43\approx0.65$, so $12.1 + 0.65\times0.1\approx12.165$). So $-\sqrt{148}\approx - 12.17$. Now, let's find the distance from -12.17 to each option:

- Distance to -11.5: $|-12.17 - (-11.5)| = |-0.67| = 0.67$
- Distance to -12.1: $|-12.17 - (-12.1)| = |-0.07| = 0.07$
- Distance to -13.4: $|-12.17 - (-13.4)| = |1.23| = 1.23$
- Distance to -13.3: $|-12.17 - (-13.3)| = |1.13| = 1.13$

The smallest distance is 0.07, which is to -12.1. Wait, but that contradicts the earlier thought. Wait, maybe I made a mistake in the square. Wait, $12^2=144$, $13^2=169$, $12.1^2=146.41$, $12.2^2=148.84$, so $148$ is 148, so $148 - 146.41 = 1.59$, $148.84 - 148 = 0.84$. So $\sqrt{148}$ is closer to 12.2? Wait, no, because 148 is 148, so 148 - 146.41 = 1.59, 148.84 - 148 = 0.84. So the distance from 148 to 146.41 is 1.59, to 148.84 is 0.84. So since 0.84 < 1.59, $\sqrt{148}$ is closer to 12.2. So $-\sqrt{148}\approx - 12.2$. Then the distance to -12.1 is $|-12.2 - (-12.1)| = 0.1$, to -13.3 is $|-12.2 - (-13.3)| = 1.1$, to -13.4 is 1.2, to -11.5 is 0.7. Wait, maybe my initial estimation was wrong. Wait, let's calculate $12.1^2 = 146.41$, $12.1^2 = 146.41$, $12.1\times12.1 = 146.41$, $12.2\times12.2 = 148.84$, so 148 is 148, so 148 - 146.41 = 1.59, 148.84 - 148 = 0.84. So the square root of 148 is 12.1657508881... So approximately 12.17. So $-\sqrt{148}\approx - 12.17$. Now, the options are -11.5, -12.1, -13.4, -13.3. So the distance from -12.17 to -12.1 is $|-12.17 + 12.1| = 0.07$. To -13.3: $|-12.17 + 13.3| = 1.13$. To -13.4: 1.23. To -11.5: 0.67. So the closest is -12.1.

Step2: Calculate the absolute differences

For each option $x$, calculate $|-\sqrt{148}-x|$:

- For $x = -11.5$: $|-\sqrt{148}+11.5| = |11.5 - \sqrt{148}|$. Since $\sqrt{148}\approx12.17$, this is $|11.5 - 12.17| = 0.67$.
- For $x = -12.1$: $|-\sqrt{148}+12.1| = |12.1 - \sqrt{148}| \approx |12.1 - 12.17| = 0.07$.
- For $x = -13.4$: $|-\sqrt{148}+13.4| = |13.4 - \sqrt{148}| \approx |13.4 - 12.17| = 1.23$.
- For $x = -13.3$: $|-\sqrt{148}+13.3| = |13.3 - \sqrt{148}| \approx |13.3 - 12.17| = 1.13$.

The smallest difference is 0.07, which corresponds to the option -12.1.

Answer:

-12.1