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.
To find the surface area of the stand, we need to find the surface area of each pedestal and add them all together.
Since each pedestal is a rectangular prism, the surface area of one pedestal is given by the formula 2lw + 2lh + 2wh, where l is the length, w is the width, and h is the height.
For pedestal 1:
l = w
w = h
h = 15
Surface area of pedestal 1 = 2lw + 2lh + 2wh
= 2(15w) + 2(15h) + 2(wh)
= 30w + 30h + 2wh
= 30w + 30h + 2w(15)
= 30w + 30(15) + 2w(15)
= 450 + 30w + 30w
= 450 + 60w
For pedestal 2:
l = w
w = h
h = 11
Surface area of pedestal 2 = 2lw + 2lh + 2wh
= 2(11w) + 2(11h) + 2(wh)
= 22w + 22h + 2wh
= 22w + 22h + 2w(11)
= 22w + 22(11) + 2w(11)
= 242 + 22w + 22w
= 242 + 44w
For pedestal 3:
l = w
w = h
h = 7
Surface area of pedestal 3 = 2lw + 2lh + 2wh
= 2(7w) + 2(7h) + 2(wh)
= 14w + 14h + 2wh
= 14w + 14h + 2w(7)
= 14w + 14(7) + 2w(7)
= 98 + 14w + 14w
= 98 + 28w
Total surface area of the stand = Surface area of pedestal 1 + surface area of pedestal 2 + surface area of pedestal 3
= (450 + 60w) + (242 + 44w) + (98 + 28w)
= 450 + 60w + 242 + 44w + 98 + 28w
= 790 + 132w
Therefore, the total surface area of the stand is 790 + 132w square units.