Which equation or inequality is true? (C) \( 3-\sqrt{9}>1 \) (D) \( \sqrt{\frac{1}{4}}>\frac{\sqrt{2}}{4} \) 17 The volume of a cube-shaped box is 216 cubic inches: What is the length of each side of the cube in inches? \( \qquad \) 8. Write the number \( 6.1 \overline{7} \) as a mixed number in simplest form. \( \qquad \) 9 Estimate the value of \( \sqrt{84} \) to the nearest tenth. \( \qquad \) 10 Solve the equation \( n^{3}=\frac{27}{343} \) for \( n \).
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For the first part, let's have some fun with these inequalities! The expression \(3 - \sqrt{9}\) simplifies to \(3 - 3\), which equals 0. That makes the first inequality false since 0 is not greater than 1. On the other hand, \(\sqrt{\frac{1}{4}} = \frac{1}{2}\), while \(\frac{\sqrt{2}}{4} \approx 0.353\), so \(0.5\) is indeed greater than \(0.353\). Thus, the second statement is true! Now, on to the cube! To find the side length, take the cube root of the volume. Since \(216 = 6^3\), the side length of the cube-shaped box is \(6\) inches. Keep that tape measure handy! For the mixed number, we convert \(6.1\overline{7}\) into a fraction. The whole number is \(6\), and the decimal .1 can be expressed as \(\frac{1}{10}\). Now, to deal with the repeating part, let’s convert \(0.\overline{7}\) into a fraction, which gives us \(\frac{7}{90}\). Summing these up, the mixed number in simplest form is \(\mathbf{6 \frac{1}{10} + \frac{7}{90} = 6 \frac{1}{9}}\). Estimation time for \(\sqrt{84}\)! We know the perfect squares near 84 are \(81\) (where \(\sqrt{81} = 9\)) and \(100\) (where \(\sqrt{100} = 10\)). A quick estimate based on this gives us a value close to \(9.2\) when you calculate it out—sneaky and simple! Lastly, for solving \(n^{3}=\frac{27}{343}\), let's break it down! Rewrite \(27\) as \(3^3\) and \(343\) as \(7^3\), leading to \(n^{3}=\left(\frac{3}{7}\right)^{3}\). Taking the cube root gives us \(n=\frac{3}{7}\). How smooth was that?
