1) The dimensions of a metal block are 2.25 m by 1.5 m by 27 cm. It is melted and recast into cubes, each of side 45 cm. How many cubes are formed?
2) If the length of each edge of a cube is doubled, how many times does its volume become? How many times does its surface area become?
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Verified answer
Answer:
[tex]\qquad \:\boxed{\begin{aligned}& \qquad \:\sf \:1) \: \: Number\:of\:cubes=10 \qquad \: \\ \\& \qquad \:\sf \:2) \: \: 8 \: times \: and \: 4 \: times \end{aligned}} \qquad \: \\ \\ [/tex]
Step-by-step explanation:
[tex]\large\underline{\sf{Solution-1}}[/tex]
Given that, a metal block of dimensions 2.25 m by 1.5 m by 27 cm is melted and recast into cubes of side 45 cm.
Length of metal block = 2.25 m = 225 cm
Breadth of metal block = 1.5 m = 150 cm
Height of metal block = 27 cm
Edge of cube = 45 cm
Let assume that number of cubes thus formed be n.
So,
[tex]\sf \: n \times Volume_{(Cube)} = Volume_{(Cuboid)} \\ \\ [/tex]
[tex]\sf \: n \times {(45)}^{3} = 225 \times 150 \times 27 \\ \\ [/tex]
[tex]\sf \: n = \dfrac{225 \times 150 \times 27}{45 \times 45 \times 45} \\ \\ [/tex]
[tex]\sf \: n = \dfrac{5 \times 150 \times 27}{1\times 45 \times 45} \\ \\ [/tex]
[tex]\sf \: n = \dfrac{5 \times 10 \times 27}{1\times 3 \times 45} \\ \\ [/tex]
[tex]\sf \: n = \dfrac{1 \times 10 \times 27}{ 3 \times 9} \\ \\ [/tex]
[tex]\sf\implies \: n = 10 \\ \\ [/tex]
Hence,
[tex]\sf\implies \:Number\:of\:cubes = 10 \\ \\ [/tex]
[tex]\large\underline{\sf{Solution-2}}[/tex]
Let assume that edge of cube be x units.
[tex]\sf\implies \sf \: Volume_{(Cube)}= {x}^{3} \\ \\ [/tex]
Now, edge of cube = 2x units.
[tex]\sf\implies \sf \: Volume_{(Cube)}'= {(2x)}^{3} = 8 {x}^{3} \\ \\ [/tex]
[tex]\sf\implies \sf \: Volume_{(Cube)}'= 8 \: Volume_{(Cube)} \\ \\ [/tex]
Now,
Let assume that edge of cube be x units.
[tex]\sf\implies \sf \: Surface\:area_{(Cube)} = 6 {x}^{2} \\ \\ [/tex]
Now, edge of cube = 2x units
[tex]\sf\implies \sf \: Surface\:area_{(Cube)}' = 6 {(2x)}^{2} = 4( {6x}^{2}) \\ \\ [/tex]
[tex]\sf\implies \sf \: Surface\:area_{(Cube)}' =4Surface\:area_{(Cube)} \\ \\ [/tex]