The sum of the perimeters of an equilateral triangle and a square is 8.Find the dimensions of the triangle and the square that produce a minimum total area.
A)side of square: ;side of triangle:
B)side of square: ;side of triangle:
C)side of square: ;side of triangle:
D)side of square: ;side of triangle:
E)side of square: ;side of triangle:

Question

The sum of the perimeters of an equilateral triangle and a square is 8.Find the dimensions of the triangle and the square that produce a minimum total area. ​
A)side of square: ;side of triangle: 
B)side of square: ;side of triangle: 
C)side of square: ;side of triangle: 
D)side of square: ;side of triangle: 
E)side of square: ;side of triangle:

Quizplus · Accepted Answer

Let x be the side length of the square and y be the side length of the equilateral triangle. Then we have:

4x + 3y = 8 (perimeter equation)
y = 2x (since the triangle is equilateral)

Substituting the second equation into the first, we get:

4x + 3(2x) = 8
10x = 8
x = 0.8

So the side length of the square is 0.8 and the side length of the equilateral triangle is 1.6. The total area is:

A = x^2 + sqrt(3)/4 * y^2
A = 0.64 + 0.6928
A = 1.3328

To minimize the total area, we need to minimize A. Taking the derivative with respect to x, we get:

dA/dx = 2x - sqrt(3)/2 * y * dy/dx
dy/dx = 0 (since y is constant)

Setting dA/dx = 0 and solving for x, we get:

x = sqrt(3)/4 * y
x = 0.866 * 2x
x = 1.732

Substituting back into the equations for x and y, we get:

x = 0.866
y = 1.732

So the best choice is B, which has a side length of 0.8 for the square and 1.732 for the equilateral triangle, giving a total area of approximately 1.3328, which is the minimum possible.

The Sum of the Perimeters of an Equilateral Triangle and a Square