Question

A rectangular solid measuring 8 cm into 4 cm into two is melted and cast the form of a cube with the side of cube is

Answer 1

${\huge{\underline{\large{\mathbb{\green{VERIFIED \: ANSWER}}}}}}$

Number Of Cubes =

$\huge \frac{Volume \: oF \: Cuboid}{Volume \: oF \: Cube} =$

$\huge \frac{8×4×2}{1×1×1} = 64$

Surface area of cuboid =2×(8×4+4×2+2×8)cm²

=2×(32+8+16)cm² = 112cm²

Surface area of 64 cubes =64×6cm²

=384 cm²

∴ Required ratio =

$\huge \frac{112}{384} = \frac{7}{24}$

= 7:24

Answer 2

$\begin{gathered}{\underline{\underline{\Huge{\dag{\textsf{\textbf{\red{Solution:}}}}}}}}\end{gathered}$

$\begin{gathered}{\underline{\underbrace{\large{\textsf{\textbf{\green{Question :}}}}}}}\end{gathered}$

• A rectangular solid measuring 8 cm into 4 cm into 2 cm is melted and cast the form of a cube. The side of the cube is?

$\begin{gathered}{\underline{\underbrace{\large{\textsf{\textbf{\green{Answer :}}}}}}}\end{gathered}$

• 4 cm

$\begin{gathered}{\underline{\underbrace{\large{\textsf{\textbf{\green{Formula \: Used :}}}}}}}\end{gathered}$

• $\sf\large\red\bigstar{Volume \: Of \: Cuboid \: = \: Volume \: Of \: Cube}$
• $\sf\large\red\bigstar{Volume \: Of \: Cuboid \: = \: Width \: \times Height \times Length}$
• $\large\red\bigstar{Volume \: of \: cube = a^{3}}$

$\begin{gathered}{\underline{\underbrace{\large{\textsf{\textbf{\green{Explanation :}}}}}}}\end{gathered}$

$\sf\large\red\bigstar{Volume \: Of \: Cuboid \: = \: Volume \: Of \: Cube}$

$\sf\leadsto{8cm \times 4cm \times 2cm \: = \: a^{3}}$

• Taking out the prime factors of 8,4,2
• Factors of 8 = 2³
• Factors of 4 = 2²
• Factors of 2 = 2×1

$\sf\leadsto \bold{ a^{3} \: = \: 2^{3} \times 2^{3}}$

• On comparing

$\sf\leadsto \bold{ a^{3} \: = \: 4^{3}}$\

$\sf\large\red\bigstar{ a \: = \: 4cm}$

$\sf\dag\underline\boxed\red{a \: = \: 4cm}$

$\begin{gathered}{\underline{\underbrace{\large{\textsf{\textbf{\green{Verification :}}}}}}}\end{gathered}$

$\sf\large\red\bigstar{Volume \: Of \: Cuboid \: = \: Volume \: Of \: Cube}$

$\sf\large\red\bigstar{Width \times \: Height \: \times Length \: = \: a^{3}}$

$\sf\large\red\bigstar{8cm \times \: 4cm \: \times \: 2cm \: = 4cm^{3}}$

$\sf\large\red\bigstar{64cm^{3} = 64cm^{3}}$

$\sf\large\blue\dag{Hence \: Verified }$

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$\begin{gathered}{\underline{\underbrace{\large{\textsf{\textbf{\green{Some \: More \: Used \: Formulas :}}}}}}}\end{gathered}$

$\begin{array}{|c|c|c|}\cline{1-3}\bf Shape&\bf Volume\ formula&\bf Surface\ area formula\\\cline{1-3}\sf Cube&\tt l^3}&\tt 6l^2\\\cline{1-3}\sf Cuboid&\tt lbh&\tt 2(lb+bh+lh)\\\cline{1-3}\sf Cylinder&\tt {\pi}r^2h&\tt 2\pi{r}(r+h)\\\cline{1-3}\sf Hollow\ cylinder&\tt \pi{h}(R^2-r^2)&\tt 2\pi{rh}+2\pi{Rh}+2\pi(R^2-r^2)\\\cline{1-3}\sf Cone&\tt 1/3\ \pi{r^2}h&\tt \pi{r}(r+s)\\\cline{1-3}\sf Sphere&\tt 4/3\ \pi{r}^3&\tt 4\pi{r}^2\\\cline{1-3}\sf Hemisphere&\tt 2/3\ \pi{r^3}&\tt 3\pi{r}^2\\\cline{1-3}\end{array}$