An edge of a cube is increased by 10%. Find the percent by which the surface area o fa cube has increased?
let each side be x units
surface area = 6x^2
new edge = 1.1x
new surface area = 6(1.1x)^2 = 7.26x^2
increase = 1.26x^2
percentage increase = 1.26x^2/(6x^2) = 1.26/6 = .21
= 21%
we could have just looked at one of the 6 faces for the same result.
To calculate the percent by which the surface area of a cube has increased when the edge length is increased by 10%, we need to understand the relationship between the edge length and the surface area of a cube.
First, let's define the formulas for the surface area and edge length of a cube:
- Surface Area of a Cube = 6 * (Edge Length)^2
- Edge Length of a Cube = (Surface Area of a Cube / 6)^(1/2)
Given that the edge of the cube has increased by 10%, we can calculate the new edge length:
New Edge Length = Old Edge Length + (10% of Old Edge Length)
= Old Edge Length + (0.10 * Old Edge Length)
= Old Edge Length * (1 + 0.10)
= Old Edge Length * 1.10
Now, let's determine the new surface area of the cube by substituting the new edge length into the surface area formula:
New Surface Area = 6 * (New Edge Length)^2
= 6 * (Old Edge Length * 1.10)^2
= 6 * (Old Edge Length^2 * 1.21)
= 6 * 1.21 * (Old Edge Length^2)
= 1.21 * (6 * Old Edge Length^2)
= 1.21 * (Old Surface Area)
Therefore, we find that the new surface area is 1.21 times the old surface area.
To calculate the percent increase, we can use the following formula:
Percent Increase = ((New Value - Old Value) / Old Value) * 100
In this case, the old surface area is the old value, and the new surface area is the new value:
Percent Increase = ((New Surface Area - Old Surface Area) / Old Surface Area) * 100
= ((1.21 * Old Surface Area - Old Surface Area) / Old Surface Area) * 100
= (0.21 * Old Surface Area / Old Surface Area) * 100
= 0.21 * 100
= 21%
Therefore, the surface area of the cube has increased by 21%.