The minimize the total area, you should use 0 meters of wire for the square and use all 16 meters for the circle.
Maximizing Total Area:
1. Define the variables:
Let x be the length of the wire used for the square.
Then, the length of the wire used for the circle is 16 - x.
2. Formulate the area equations:
Area of the square: A_s = x^2 / 4
Area of the circle: A_c = πr^2, where r is the radius of the circle.
3. Relate the variables:
The circumference of the circle is equal to the perimeter of the square: 4x = 2πr.
Solve for r: r = 2x / π.
4. Substitute r in the area equation for the circle:
A_c = π(2x/π)^2 = 4x^2 / π
5. Formulate the total area equation:
Total area: A_t = A_s + A_c = x^2 / 4 + 4x^2 / π
6. Find the maximum total area:
Take the derivative of A_t with respect to x and set it equal to zero: dA_t/dx = (x / 2) - (8x / π) = 0.
Solve for x: x = 4π / 3.
Verify that it's a maximum by checking the second derivative.
7. Use the value of x to find the length for the square:
Square wire length: x = 4π / 3 = 4.19 m (approximately).
Therefore, to maximize the total area, you should use approximately 4.19 meters of wire for the square.
Minimizing Total Area:
1. Define the variables:
Use the same definitions as above.
2. Formulate the area equations:
Use the same equations as above.
3. Relate the variables:
Use the same equation as above.
4. Substitute r in the area equation for the circle:
Use the same equation as above.
5. Formulate the total area equation:
Use the same equation as above.
6. Find the minimum total area:
Since the quadratic term in A_t is always positive, the minimum total area occurs when x is minimized. Thus, you should use the minimum possible value for x, which is:
Minimum square wire length: x = 0 m.
Therefore, to minimize the total area, you should use 0 meters of wire for the square and use all 16 meters for the circle.