anonymous
  • anonymous
HELP! PIC BELOS I GIVE MEDALS:) the lateral area of the right regular triangular prism is 18pi cm^2. find the total surface area. give your answer in decimal form rounded of to 2 decimal places
Geometry
schrodinger
  • schrodinger
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anonymous
  • anonymous
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mathstudent55
  • mathstudent55
The lateral area of the prism is the sum of the areas of the three rectangles whose dimensions are 3 and x.
mathstudent55
  • mathstudent55
Each rectangular face has an area of 3x, so all three faces have a combined area of 3 * 3x = 9x. The lateral area = 9x = 18pi

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mathstudent55
  • mathstudent55
9x = 18pi x = 2pi
mathstudent55
  • mathstudent55
Now we need the area of the bases.
mathstudent55
  • mathstudent55
Each base is a regular triangle with side = 2pi
mathstudent55
  • mathstudent55
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mathstudent55
  • mathstudent55
Each base of the prism is a regular (equilateral) triangle with side length of 2pi. If you drop an altitude, as the picture above shows, you create two 30-60-90 triangles. The ratio of the lengths of the legs of a 30-60-90 triangle is 1 : sqrt(3) Since the short leg measures pi, the long leg measures pi * sqrt(3) The long leg is the altitude of the large triangle, whose base measure 2pi
mathstudent55
  • mathstudent55
The area of a triangle is A = (1/2)bh Here we have A = (1/2)(2pi)(pi * sqrt(3) ) = pi^2*sqrt(3) Since there are two bases, the area of the 2 bases is 2*pi^2*sqrt(3)
mathstudent55
  • mathstudent55
Since the lateral area is 18pi, the total surface area is SA = (18pi + 2*pi^2*sqrt(3) ) cm^2 Now you can calculate an approximate value with your calculator.
anonymous
  • anonymous
72.21
mathstudent55
  • mathstudent55
I get 90.67

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