At the Olympics, the winners of the gold, silver, and bronze medals stand on a tiered pedestal.
As labeled in the diagram, the height of pedestal 3 is 7 inches, the height of pedestal 2 is 11 inches, and the height of pedestal 1 is 15 inches. Assuming each pedestal is of equal width and depth, find the total surface area of the stand.
(1 point)
To find the total surface area of the stand, we first need to find the surface area of each individual pedestal.
Since each pedestal is a rectangular prism, the formula for the surface area of a rectangular prism is:
Surface Area = 2lw + 2lh + 2wh
where l is the length, w is the width, and h is the height of the prism.
For pedestal 3:
Length = Width = x (let's call the width and length of the pedestal x)
Height = 7 inches
Surface Area of pedestal 3 = 2(x)(x) + 2(x)(7) + 2(7)(x)
Surface Area of pedestal 3 = 2x^2 + 14x + 14x
Surface Area of pedestal 3 = 2x^2 + 28x
For pedestal 2:
Length = Width = x
Height = 11 inches
Surface Area of pedestal 2 = 2(x)(x) + 2(x)(11) + 2(11)(x)
Surface Area of pedestal 2 = 2x^2 + 22x + 22x
Surface Area of pedestal 2 = 2x^2 + 44x
For pedestal 1:
Length = Width = x
Height = 15 inches
Surface Area of pedestal 1 = 2(x)(x) + 2(x)(15) + 2(15)(x)
Surface Area of pedestal 1 = 2x^2 + 30x + 30x
Surface Area of pedestal 1 = 2x^2 + 60x
Now, the total surface area of the stand is the sum of the surface areas of all three pedestals:
Total Surface Area = Surface Area of pedestal 3 + Surface Area of pedestal 2 + Surface Area of pedestal 1
Total Surface Area = (2x^2 + 28x) + (2x^2 + 44x) + (2x^2 + 60x)
Total Surface Area = 6x^2 + 132x
Therefore, the total surface area of the stand is 6x^2 + 132x square inches.