The combinated permeter of a circle and a squar is 16 find the dimensions of the circle and the square that produces a minimum total area?

To find the dimensions of the circle and square that produce a minimum total area, we need to set up an equation and solve it using calculus. Let's denote the radius of the circle as r and the side length of the square as s.

The perimeter of a circle is given by P_circle = 2πr, and the perimeter of a square is P_square = 4s. We are given that the combined perimeter is 16, so we can set up the equation:

P_circle + P_square = 16

Substituting the formulas for the perimeters:

2πr + 4s = 16

To find the minimum total area, we need to express the area as a function of a single variable. The area of a circle is given by A_circle = πr^2, and the area of a square is A_square = s^2. The total area is the sum of the areas:

A_total = A_circle + A_square

Substituting the formulas for the areas:

A_total = πr^2 + s^2

To solve for the minimum total area, we need to find the values of r and s that minimize A_total. We can do this by finding the critical points of A_total with respect to r and s.

Taking the partial derivative of A_total with respect to r:

∂A_total/∂r = 2πr

Taking the partial derivative of A_total with respect to s:

∂A_total/∂s = 2s

Setting both derivatives to zero to find the critical points:

2πr = 0 (equation 1)
2s = 0 (equation 2)

From equation 2, we can see that s = 0, but that doesn't make sense since we are looking for dimensions. Therefore, we ignore s = 0.

From equation 1, we get r = 0. We can also observe that as r gets larger, A_total increases, so there is no minimum point when r = 0.

Next, we need to consider the boundary of the problem. Since r represents the radius of the circle, it cannot be negative. So we only need to consider the positive values of r.

Now, we need to substitute the value of r from equation 1 into the perimeter equation:

2πr + 4s = 16
2π(0) + 4s = 16
4s = 16
s = 4

So the solution is r = 0 (ignored), s = 4.

Therefore, the dimensions of the circle and square that produce a minimum total area are a circle with radius 0 (which is not possible) and a square with side length 4.