If the length of the longest rod that can be kept in a hollow cube increases by 10%, what will be the percentage increase in the volume of the cube?
12.1%
13%
30%
33.1%
Answers
Answer:
33.1%
Step-by-step explanation:
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The percentage increase in the volume of the cube is equal to 33.1 %. (Option-4)
Given,
The length of the longest rod that can be kept in a hollow cube increases by 10%.
To find,
The percentage increase in the volume of the cube.
Solution,
We can simply solve this mathematical problem using the following process:
Let us assume that the length of each side of the given hollow cube is "a" unit.
Mathematically,
For a cube having each equal side of length "a" units;
The length of the longest rod that can be kept in the hollow cube
= the length of the longest diagonal of the cube
= (√3a) units
= √3 × (length of each side)
{Statement-1}
The volume of a cube = (length of each side)^3
{Statement-2}
Now, according to the question and statement-1;
The length of the longest rod that can be kept in the hollow cube in the initial condition
= (√3a) units
Now, when the length of the longest rod increases by 10%, the final length becomes
= (10/100)×(√3a) units + (√3a) units
= (√3a/10) units + (√3a) units
= (11√3a/10) units
= (11a/10)×√3 units
=> length of each side, when the length of the longest rod increases by 10%, is = (11a/10) units
{according to statement-1}
Now, according to statement-2;
The volume of the cube in the initial conditions
= (initial length of each side)^3
= a^3 cubic units
And, the volume of the cube in the final conditions
= (final length of each side)^3
= (11a/10)^3 cubic units
= (1331a/1000) cubic units
Now, the percentage increase in the volume of the cube
= (change in volume)/(initial Volume)×100
= {(1331a^3/1000) - (a^3)}/(a^3) × 100
= (1331-1000)/1000 × 100
= 331/10% = 33.1 %
Hence, the percentage increase in the volume of the cube is equal to 33.1 %. (Option-4)