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The dimensions of a pyramid are increased by 1100%. The percentage increase in surface area is
The new dimensions are ___times the original dimensions, and the new surface area is ___times the original surface area. The increase in surface area is___%.
the percent increase is 14300%
What's the percent increace?
To find the percentage increase in surface area, we first need to determine the new dimensions of the pyramid.
Let's assume the original dimensions of the pyramid are "x" units for length, "y" units for width, and "z" units for height.
When the dimensions are increased by 1100%, it means they are multiplied by 1 + (1100/100) = 12.
So, the new dimensions would be 12x units for length, 12y units for width, and 12z units for height.
Next, let's calculate the surface areas.
The surface area of a pyramid is given by the formula:
Surface Area = (1/2) x base x height + (length x width)
The original surface area = (1/2) x x x z + (x x y)
= (1/2)xz + xy
= (1/2)(xz + 2xy)
The new surface area = (1/2) x (12x) x (12z) + (12x)(12y)
= (1/2) x (12)(12xz + 12xy)
= (1/2) x 12^2(xz + 2xy)
= 72(xz + 2xy)
To find the percentage increase in surface area, we can use the formula:
Percentage Increase = [(New Value - Original Value) / Original Value] x 100
Let's substitute the values into the formula:
Percentage Increase = [(72(xz + 2xy) - (1/2)(xz + 2xy)) / (1/2)(xz + 2xy)] x 100
= [(144(xz + 2xy) - (xz + 2xy)) / (xz + 2xy)] x 100
= [(143(xz + 2xy)) / (xz + 2xy)] x 100
= 14300
Therefore, the increase in surface area is 14300%.
The new dimensions are 12 times the original dimensions, and the new surface area is 143 times the original surface area.
The surface areas of similar objects are proportional to the square of their corresponding dimensions.
So if one dimension of the original is x
then the corresponding dimension of the new object is 1100% of x
= 11x
surface area of old/surface area of new = x^2/(11x^2) = 1/121
the new surface area is 121 times as large as the original