The combined perimeter of an equilateral triangle and square is 13. Find the dim
ID: 2859066 • Letter: T
Question
The combined perimeter of an equilateral triangle and square is 13.Find the dimensions of the triangle and square that produce a minimum total area. The square measures ______ on each side. The triangle measures _______ on each side.
Find the dimensions of the triangle and square that produce a maximum total area. The square measures _____ on each side. The triangle measures ______ on each side.
The combined perimeter of an equilateral triangle and square is 13.
Find the dimensions of the triangle and square that produce a minimum total area. The square measures ______ on each side. The triangle measures _______ on each side.
Find the dimensions of the triangle and square that produce a maximum total area. The square measures _____ on each side. The triangle measures ______ on each side.
Find the dimensions of the triangle and square that produce a minimum total area. The square measures ______ on each side. The triangle measures _______ on each side.
Find the dimensions of the triangle and square that produce a maximum total area. The square measures _____ on each side. The triangle measures ______ on each side.
Explanation / Answer
let s be the side of square and a be side of equilateral triangle.
Combined perimeter = 13
==> 4s + 3a = 13
==> 3a = 13 - 4s
==> a = (13 - 4s)/3
Area A = s2 + (3 /4) a2
==> A(s) = s2 + (3 /4) ((13 - 4s)/3)2
==> A(s) = s2 + (3 /36) (169 - 104s + 16s2)
==> A(s) = s2 + (1693 /36) - (1043 /36)s + (163 /36)s2)
==> A(s) = ((9 + 43)/9 )s2 - (1043 /36)s
critical points ==> A '(s) = 0
==> ((9 + 43)/9 ) (2)s- (1043 /36) = 0
==> 3.539601s - 5.003702 = 0
==> s = 5.003702/3.539601
==> s = 1.413631
==> 4(1.413631) + 3a = 13
==> 3a = 13 - 5.654524
==> 3a = 7.345476
==> a = 2.44849
A ''(s) = ((9 + 43)/9 ) (2)(1) - 0 = ((9 + 43)/9 ) (2) > 0
Hence at square side s = 1.415 , equlateral triangle a = 2.448 , the Area is minimum
Maximum occurs at extreme points
2 cases are side of square = 0 , side of equliateral triangle = 13/3
and side of square = 13/4 , side of equliateral triangle = 0
Area A1 = (3 /4) (13/3)2 = 8.131
A2 = (13/4)2 = 10.563
Hence maximum area at side of square = 13/4 and side of triangle = 0