if each edge of a cube is doubled then the percentage increase in its total surface area is?
Answers
To get the surface area of a cube, we use the formula:
6s^2
where "s" is the length of any of the edges of the cube. This is due to the fact that all edges of a cube are of equal length. What this formula does is first, calculate the area of one face, which is multiplying the length of two edges: s x s or simply, s^2. Since a cube has six equal faces (back, front, two sides, bottom, top), we multiply the area of a face by 6 to get the total surface area.
Increasing the length of a cube's edges by a factor of 2 can be written as:
2s
This is now the new length of each edges. The surface area will now be equal to:
6(2s)^2
Distributing the exponent we get:
6(4s^2)
Further simplifying:
24s^2
This now gives the total surface area relative to the original length of the edges "s".
Subtracting the original total surface area with the new one we get:
24s^2 - 6s^2 = 18s^2
There's an increase of 18s^2, which is THREE times 6s^2.
Therefore, there is a 300% increase in the total surface area.
Define x:
Let x be the length of the edge
⇒ length = x
⇒ Surface area = 6x²
When the edge is doubled
⇒ length = 2x
⇒ Surface area = 6(2x)² = 24x²
Find the percentage increased in surface area:
Increase = 24x² - 6x² = 18x²
Percentage increase = increase ÷ original x 100
Percentage increase = (18x² ÷ 6x²) x 100 = 300%
Answer: The surface area is increased 300%
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ALTERNATIVE METHOD (SHORT CUT) :
Increase in area = (increase in length)²
Length is double ⇒ Length is increased two times
New area = (2)² = 4 times
Difference = 4 - 1 = 3 times = 300%
Answer: The surface area is increased 300%