Question

four rational approximations of the irrational number \\(\sqrt{62}\\) are \\(7\frac{17}{20}\\), \\(7.8\overline{7}\\), \\(7.87\\), and \\(7.9\\). determine which approximation is closest to \\(\sqrt{62}\\) without going over. answer questions 1–4 below.

1. estimate \\(\sqrt{62}\\) using perfect squares. between which two whole numbers is \\(\sqrt{62}\\) located?

\\(\sqrt{62}\\) is between two whole numbers. the smaller whole number is \\(\square\\) and the larger whole number is \\(\square\\).
(type whole numbers.)
narrow the interval by estimating \\(\sqrt{62}\\).
\\(\sqrt{62} \approx \square\\) (round to the nearest tenth as needed.)

1. complete the table by writing the numbers in decimal form.

estimate	decimal form
\\(7.8\overline{7}\\)	\\(\square\\)
\\(7.87\\)	\\(\square\\)

(round to the nearest hundredth as needed.)

1. label the tick marks on the number line and plot the three estimates from the table above.

Explanation:

Step1: Find whole number bounds

Identify perfect squares around 62: $7^2=49$, $8^2=64$. Since $49<62<64$, $\sqrt{62}$ is between 7 and 8.

Step2: Estimate $\sqrt{62}$ to tenth

Calculate $7.9^2=62.41$, $7.8^2=60.84$. 62 is closer to 62.41, so $\sqrt{62}\approx7.9$.

Step3: Convert $7\frac{17}{20}$ to decimal

Calculate $\frac{17}{20}=0.85$, so $7+0.85=7.85$.

Step4: Convert $7.8\overline{7}$ to decimal

$7.8\overline{7}=7.8777...$, round to hundredth: 7.88.

Step5: 7.87 in decimal form

7.87 is already in decimal form, so it stays 7.87.

Answer:

1. Smaller whole number: 7; Larger whole number: 8; $\sqrt{62} \approx 7.9$
2. $7\frac{17}{20}$: 7.85; $7.8\overline{7}$: 7.88; 7.87: 7.87