Question

a 30 inch piece of string is to be cut into two pieces. the first piece will be formed into the shape of an equilateral triangle and the second piece into a square. Find the length of the first piece if the combined area of the triangle and the square is to be as small as possible.

Answer 1

To minimize the combined area of the triangle and the square, we can calculate the length of the first piece of string for the equilateral triangle using calculus. The formula for the area of an equilateral triangle and square can be used to find the combined area based on the length of the first piece of string. By minimizing this combined area, we can determine the length of the first piece of string that should be used.

Step-by-step explanation:

To find the length of the first piece of string that should be formed into the shape of an equilateral triangle to minimize the combined area of the triangle and the square, we can follow these steps:

1. Let x be the length of the first piece of string for the equilateral triangle.
2. The perimeter of the equilateral triangle is 3x, so each side of the triangle is x/3.
3. The area of an equilateral triangle is given by the formula A = (sqrt(3)/4) * s^2, where s is the length of a side. In this case, the area of the triangle is A = (sqrt(3)/4) * (x/3)^2.
4. The length of the second piece of string for the square is 30 - x.
5. The area of a square is given by the formula A = s^2, where s is the length of a side. In this case, the area of the square is A = (30 - x)^2.
6. The combined area of the triangle and the square is given by the sum of their areas: (sqrt(3)/4) * (x/3)^2 + (30 - x)^2.
7. To minimize this combined area, we can take the derivative with respect to x, set it equal to zero, and solve for x.
8. After finding the value of x, we can calculate the length of the first piece of string for the equilateral triangle.