Question

which number is closest to $sqrt{65}$?
a. 8
b. 9
c. 32
d. 64

Explanation:

Step1: Find perfect squares around 65

We know that \(8^2 = 64\) and \(9^2 = 81\). So, \(\sqrt{64}=8\) and \(\sqrt{81}=9\).

Step2: Compare distances from 65

The distance between 65 and 64 is \(|65 - 64| = 1\). The distance between 65 and 81 is \(|65 - 81| = 16\). Since 1 < 16, 65 is closer to 64. So, \(\sqrt{65}\) is closer to \(\sqrt{64}=8\)? Wait, no, wait. Wait, 8^2=64, 9^2=81. Wait, 65 - 64 = 1, 81 - 65 = 16. So 65 is closer to 64, so \(\sqrt{65}\) is closer to \(\sqrt{64}=8\)? Wait, no, wait, 8^2=64, 9^2=81. So \(\sqrt{65}\) is between 8 and 9. Now, 65 - 64 = 1, 81 - 65 = 16. So the difference between 65 and 64 is 1, which is smaller than the difference between 81 and 65 (16). So \(\sqrt{65}\) is closer to 8? Wait, no, wait, 8 is \(\sqrt{64}\), 9 is \(\sqrt{81}\). So \(\sqrt{65}\) is 8.062..., so it's closer to 8? But wait, the options are A.8, B.9, C.32, D.64. Wait, no, wait, maybe I made a mistake. Wait, 8^2=64, 9^2=81. So \(\sqrt{65}\) is approximately 8.06, which is closer to 8? But wait, let's check again. 8.06 - 8 = 0.06, 9 - 8.06 = 0.94. So 0.06 < 0.94, so it's closer to 8? But wait, the options: A.8, B.9, C.32, D.64. Wait, but maybe I misread. Wait, the question is which number is closest to \(\sqrt{65}\). So \(\sqrt{65}\approx8.06\). So the closest number among the options is 8 (A) or 9 (B). Since 8.06 - 8 = 0.06 and 9 - 8.06 = 0.94, so 8 is closer. But wait, wait, maybe I made a mistake. Wait, 8^2=64, 9^2=81. So 65 is 1 more than 64, so \(\sqrt{65}\) is 8 + 1/(28) approximately (using linear approximation: \(f(x)=\sqrt{x}\), \(f'(x)=1/(2\sqrt{x})\), at x=64, f(64)=8, f'(64)=1/16. So f(65)=f(64)+f'(64)(65 - 64)=8 + 1/16≈8.0625). So it's approximately 8.06, so closer to 8. But wait, the options: A.8, B.9, C.32, D.64. So the answer should be A? Wait, but maybe I misread the options. Wait, the options are A.8, B.9, C.32, D.64. So yes, 8 is closer to \(\sqrt{65}\) (≈8.06) than 9 (9 - 8.06 = 0.94), and 32 and 64 are way off. So the answer is A.8? Wait, but let's check again. 8.06 is closer to 8 than to 9. So yes.

Answer:

A. 8