SOLUTION: The apothem of a regular polygon is the perpendicular distance from the center of the polygon to a side. The area, A, of a regular polygon varies jointly as the apothem, a, and the

Algebra ->  Test -> SOLUTION: The apothem of a regular polygon is the perpendicular distance from the center of the polygon to a side. The area, A, of a regular polygon varies jointly as the apothem, a, and the      Log On


   

Question 33408: The apothem of a regular polygon is the perpendicular distance from the center of the polygon to a side. The area, A, of a regular polygon varies jointly as the apothem, a, and the perimeter,p. A regular triangle with an apothem of 3 inches and a perimeter of 31.2 inches has an area of 46.8 square inches.
Find the constant variation
Write a joint variation equation
Find the area of a regular triangle with an apothem of 2.3 inches and a perimeter of 12 inches.

Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
First, you can write:
A+=+%28k%29%28a%29%28p%29
To find the value of k, the constant of variation, substitute the given values of A, a, and p.
46.8+=+%28k%29%283%29%2831.2%29 Solve for k.
46.8+=+93.6%28k%29 Divide both sides by 93.6
46.8%2F93.6+=+k
k+=+0.5 Constant of variation.
A+=+0.5ap Joint variation equation.
A+=+0.5%282.3%29%2812%29
A+=+13.8sq.ins.
There is a problem with answer!
If you calculate the area of the given regular (equilateral) triangle using Heron's formulaA+=+sqrt%28s%28s-a%29%28s-b%29%28s-c%29%29, you will get 6.9 sq.ins. which is just half of what I got using the direct variation method above.
I have concluded that the given apothem (a = 2.3 inches) is just twice what it should be for a regular triangle of 4 inches per side.
You might want to check this out.