Question

problem 14
tell which of these numbers is greater. (use an estimate for the irrational.)
after you enter your answer press go.
a. \\(\sqrt{18}\\) b. 4.75

Explanation:

Step1: Estimate $\sqrt{18}$

We know that $4^2 = 16$ and $5^2 = 25$. Since $16<18<25$, $\sqrt{16}<\sqrt{18}<\sqrt{25}$, so $4<\sqrt{18}<5$. Also, $4.2^2 = 17.64$ and $4.3^2 = 18.49$. Since $17.64<18<18.49$, $\sqrt{18}$ is between $4.2$ and $4.3$. A better estimate: $4.24^2 = 17.9776$ and $4.25^2 = 18.0625$. So $\sqrt{18}\approx4.24$ (more accurately, around $4.2426$).

Step2: Compare with 4.75

Now we compare the estimated value of $\sqrt{18}$ (≈4.24) with 4.75. Since $4.24<4.75$, we can see that $4.75$ is greater than $\sqrt{18}$. Wait, no, wait: Wait, $4.24^2 = 17.9776$, $4.25^2 = 18.0625$. So $\sqrt{18}\approx4.2426$. Now compare with 4.75. Since $4.2426 < 4.75$? Wait, no, wait, 4.24 is less than 4.75? Wait, no, 4.24 is less than 4.75? Wait, 4.24 is 4 and 0.24, 4.75 is 4 and 0.75. So 0.24 < 0.75, so 4.24 < 4.75? Wait, no, wait, I made a mistake. Wait, $4.24^2 = 17.9776$, $4.25^2 = 18.0625$. So $\sqrt{18}\approx4.2426$. Now 4.2426 vs 4.75. 4.2426 is less than 4.75? Wait, no, 4.24 is less than 4.75? Wait, 4.24 < 4.75? Yes, because the tenths place: 2 < 7. So 4.24 < 4.75? Wait, but wait, maybe I miscalculated. Wait, no, 4.24 is 4.24, 4.75 is 4.75. So 4.24 < 4.75. Wait, but that would mean B is greater. But wait, let's check again. Wait, $\sqrt{16}=4$, $\sqrt{25}=5$. $\sqrt{18}$ is between 4 and 5. 4.75 is also between 4 and 5. Let's square 4.75 to see. $4.75^2=(4 + 0.75)^2=4^2 + 2*4*0.75 + 0.75^2=16 + 6 + 0.5625=22.5625$. Wait, no, that's wrong. Wait, no, $4.75^2$: 4.75 4.75. Let's calculate: 4 4 = 16, 4 0.75 = 3, 0.75 4 = 3, 0.75 0.75 = 0.5625. So (4 + 0.75)(4 + 0.75) = 16 + 3 + 3 + 0.5625 = 22.5625. Wait, that's way bigger than 18. Wait, no, I messed up. Wait, the number is 4.75, not 47.5. Wait, 4.75 squared: 4.75 4.75. Let's do 475 * 475 = 225625, then divide by 10000 (since 4.75 = 475/100), so 225625 / 10000 = 22.5625. Wait, that's correct. But $\sqrt{18}$ is about 4.24. So 4.24 < 4.75? Wait, no, 4.24 is less than 4.75? Yes, because 4.24 is 4 and 0.24, 4.75 is 4 and 0.75. So 0.24 < 0.75, so 4.24 < 4.75. Wait, but that would mean B is greater. But wait, maybe I made a mistake in estimation. Wait, $\sqrt{18}$: 4.2426, 4.75 is 4.75. So 4.2426 < 4.75. So B is greater. Wait, but let's check again. Wait, 4.2426 squared is 18, 4.75 squared is 22.5625. So yes, 4.75 is greater than $\sqrt{18}$. Wait, but the options are A: $\sqrt{18}$, B: 4.75. So which is greater? B is greater. Wait, but maybe I messed up the estimation. Wait, no, 4.24^2 = 17.9776, 4.25^2 = 18.0625. So $\sqrt{18}\approx4.2426$. 4.2426 < 4.75. So B is greater.

Answer:

B. 4.75