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If the length, breadth and height of a cube are decreased, decreased and increased by 5%, 5% and 20%, respectively, then what will be the impact on the surface area of the cube (in percentage terms)?
- a)6.0833%
- b)5%
- c)8.33%
- d)7.25%
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
If the length, breadth and height of a cube are decreased, decreased a...
Let the initial side of the cube be 'x' units.
⇒ Surface area of cube =6×side2=6x2 square units
Given, the length, breadth and height of a cube are decreased, decreased and increased by 5%, 5% and 20% respectively.
This means that the cube is now converted into a cuboid.
Length of cuboid =(100-5)% of x=95% of x=19x/20
Breadth of cuboid =(100-5)% of x=95% of x=19x/20
Height of cuboid =(100+20)% of x=120% of x=24x/20
⇒ Surface area of cuboid =2[(19x/20×19x/20)+(19x/20×24x/20)+(24x/20×19x/20)]
=2[(361x2/400)+(456x2/400)+(456x2/400)]
=2×[1273x2/400]=1273x2/200 square units
⇒ Increase in surface area ={[(1273x2/200)-6x2] / [6x2]}×100=(73/1200)×100=6.0833%
Hence, the correct answer is 6.0833%.
This question is part of UPSC exam. View all Quant courses
Most Upvoted Answer
If the length, breadth and height of a cube are decreased, decreased a...
Given: The length, breadth and height of a cube are decreased, decreased and increased by 5%, 5% and 20%, respectively.
To find: The impact on the surface area of the cube (in percentage terms).
Let's assume the initial length, breadth and height of the cube to be L, B and H respectively.
After the given changes, the new length, breadth and height of the cube will be:
New length = 0.95L (decreased by 5%)
New breadth = 0.95B (decreased by 5%)
New height = 1.2H (increased by 20%)
Formula for the surface area of a cube is 6a^2, where a is the length of one of its edges.
Initial surface area of the cube = 6L^2 (as all edges are equal)
New surface area of the cube = 6(0.95L)^2 + 6(0.95B)^2 + 6(1.2H)^2
= 6(0.9025L^2) + 6(0.9025B^2) + 6(1.44H^2)
= 6(0.9025(L^2 + B^2) + 1.44H^2)
Let's calculate the percentage change in surface area by using the formula:
Percentage Change = [(New Value - Old Value)/Old Value] x 100
Percentage Change in surface area = [(6(0.9025(L^2 + B^2) + 1.44H^2) - 6L^2)/6L^2] x 100
Simplifying the above equation, we get:
Percentage Change = 6.0833%
Therefore, the impact on the surface area of the cube is a decrease of 6.0833%. Hence, the correct answer is option (a).
To find: The impact on the surface area of the cube (in percentage terms).
Let's assume the initial length, breadth and height of the cube to be L, B and H respectively.
After the given changes, the new length, breadth and height of the cube will be:
New length = 0.95L (decreased by 5%)
New breadth = 0.95B (decreased by 5%)
New height = 1.2H (increased by 20%)
Formula for the surface area of a cube is 6a^2, where a is the length of one of its edges.
Initial surface area of the cube = 6L^2 (as all edges are equal)
New surface area of the cube = 6(0.95L)^2 + 6(0.95B)^2 + 6(1.2H)^2
= 6(0.9025L^2) + 6(0.9025B^2) + 6(1.44H^2)
= 6(0.9025(L^2 + B^2) + 1.44H^2)
Let's calculate the percentage change in surface area by using the formula:
Percentage Change = [(New Value - Old Value)/Old Value] x 100
Percentage Change in surface area = [(6(0.9025(L^2 + B^2) + 1.44H^2) - 6L^2)/6L^2] x 100
Simplifying the above equation, we get:
Percentage Change = 6.0833%
Therefore, the impact on the surface area of the cube is a decrease of 6.0833%. Hence, the correct answer is option (a).
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