Answer
Verified
412.2k+ views
Hint: The ratio of the volume of the cubes can be used to calculate the ratio of the sides. After obtaining the ratio of sides, the ratio of surface area can be easily determined.
Complete step-by-step answer:
Given in the problem, two cubes have their volumes in the ratio 1:27.
We need to find the ratio of their surface areas.
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.
If a cube has a side $x {\text { unit}}$, then the volume of the cube is given by ${x^3}{\left( \text {unit} \right)^3}$.
Also, the surface area of the cube is given by $6{x^2}{\left( \text {unit} \right)^2}$.
In the problem there are two cubes given.
Let the side of the first cube be $x{\text{ }}unit$ and that of the second be $y{\text{ }}unit$ respectively.
Hence, the volume of the first cube will be ${x^3}{\left( \text{unit} \right)^3}$.
And the volume of the second cube will be ${y^3}{\left( \text {unit} \right)^3}$.
It is given that the ratio of their volumes is 1:27.
\[
\Rightarrow \dfrac{{{\text{Volume of first cube}}}}{{{\text{Volume of second cube}}}} = \dfrac{1}{{27}} \\
\Rightarrow \dfrac{{{x^3}}}{{{y^3}}} = {\left( {\dfrac{x}{y}} \right)^3} = \dfrac{1}{{27}} \\
\Rightarrow \dfrac{x}{y} = {\left( {\dfrac{1}{{27}}} \right)^{\dfrac{1}{3}}} = \dfrac{1}{3} ………....(1) \\
\]
Hence the ratio of the side of the first cube to that of the second cube is 1:3.
We need to find the ratio of their surface area.
By using the above-mentioned formula, we get,
Surface area of the first cube $ = 6{x^2}{\left( \text {unit} \right)^2}$
Surface area of the second cube $ = 6{y^2}{\left( \text {unit} \right)^2}$
Therefore, ratio of their surface area is given by
\[ \Rightarrow \dfrac{{{\text{Surface area of first cube}}}}{{{\text{Surface area of second cube}}}} = \dfrac{{6{x^2}}}{{6{y^2}}} = {\left( {\dfrac{x}{y}} \right)^2}\]
Using equation (1) in the above, we get
\[ \Rightarrow \dfrac{{{\text{Surface area of first cube}}}}{{{\text{Surface area of second cube}}}} = {\left( {\dfrac{x}{y}} \right)^2} = {\left( {\dfrac{1}{3}} \right)^2} = \dfrac{1}{9}\]
Hence the ratio of surface area of the first cube to that of the second cube is 1:9.
Therefore, option (C). 1:9 is the correct answer.
Note: The formula of volume and surface area of the cube should be kept in mind while solving problems like above. Ratio is the quantitative relation between two amounts showing the number of times one value contains or is contained within the other. The unit of both the quantities in the ratio should be the same. In problems like the above effort should be made to obtain the desired result while assuming the minimum number of unknown quantities.
Complete step-by-step answer:
Given in the problem, two cubes have their volumes in the ratio 1:27.
We need to find the ratio of their surface areas.
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.
If a cube has a side $x {\text { unit}}$, then the volume of the cube is given by ${x^3}{\left( \text {unit} \right)^3}$.
Also, the surface area of the cube is given by $6{x^2}{\left( \text {unit} \right)^2}$.
In the problem there are two cubes given.
Let the side of the first cube be $x{\text{ }}unit$ and that of the second be $y{\text{ }}unit$ respectively.
Hence, the volume of the first cube will be ${x^3}{\left( \text{unit} \right)^3}$.
And the volume of the second cube will be ${y^3}{\left( \text {unit} \right)^3}$.
It is given that the ratio of their volumes is 1:27.
\[
\Rightarrow \dfrac{{{\text{Volume of first cube}}}}{{{\text{Volume of second cube}}}} = \dfrac{1}{{27}} \\
\Rightarrow \dfrac{{{x^3}}}{{{y^3}}} = {\left( {\dfrac{x}{y}} \right)^3} = \dfrac{1}{{27}} \\
\Rightarrow \dfrac{x}{y} = {\left( {\dfrac{1}{{27}}} \right)^{\dfrac{1}{3}}} = \dfrac{1}{3} ………....(1) \\
\]
Hence the ratio of the side of the first cube to that of the second cube is 1:3.
We need to find the ratio of their surface area.
By using the above-mentioned formula, we get,
Surface area of the first cube $ = 6{x^2}{\left( \text {unit} \right)^2}$
Surface area of the second cube $ = 6{y^2}{\left( \text {unit} \right)^2}$
Therefore, ratio of their surface area is given by
\[ \Rightarrow \dfrac{{{\text{Surface area of first cube}}}}{{{\text{Surface area of second cube}}}} = \dfrac{{6{x^2}}}{{6{y^2}}} = {\left( {\dfrac{x}{y}} \right)^2}\]
Using equation (1) in the above, we get
\[ \Rightarrow \dfrac{{{\text{Surface area of first cube}}}}{{{\text{Surface area of second cube}}}} = {\left( {\dfrac{x}{y}} \right)^2} = {\left( {\dfrac{1}{3}} \right)^2} = \dfrac{1}{9}\]
Hence the ratio of surface area of the first cube to that of the second cube is 1:9.
Therefore, option (C). 1:9 is the correct answer.
Note: The formula of volume and surface area of the cube should be kept in mind while solving problems like above. Ratio is the quantitative relation between two amounts showing the number of times one value contains or is contained within the other. The unit of both the quantities in the ratio should be the same. In problems like the above effort should be made to obtain the desired result while assuming the minimum number of unknown quantities.
Recently Updated Pages
Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred
The branch of science which deals with nature and natural class 10 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Define absolute refractive index of a medium
Find out what do the algal bloom and redtides sign class 10 biology CBSE
Prove that the function fleft x right xn is continuous class 12 maths CBSE
Trending doubts
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE
Which type of bond is stronger ionic or covalent class 12 chemistry CBSE
What organs are located on the left side of your body class 11 biology CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
How fast is 60 miles per hour in kilometres per ho class 10 maths CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
When people say No pun intended what does that mea class 8 english CBSE